Additive and geometric transversality of fractal sets in the integers

By juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of ×r$\times r$ ‐invariant sets on the 1‐torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include: a classification of all subsets of the positive integers that are simultaneously ×r$\times r$ ‐ and ×s$\times s$ ‐invariant; integer analogs of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection A∩B$A\cap B$ and the sumset A+B$A+B$ of ×r$\times r$ ‐ and ×s$\times s$ ‐invariant sets A$A$ and B$B$ when r$r$ and s$s$ are multiplicatively independent; and a description of the dimension of iterated sumsets A+A+⋯+A$A+A+\cdots +A$ for any ×r$\times r$ ‐invariant set A$A$ . We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the Lq$L^q$ ‐dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.


Introduction
Number theorists in the first half of the 20th century were among the first to consider the degree to which base 2 and base 3 representations of real numbers are independent. An open conjecture attributed to Mahler [MF] postulates, for example, that if (a n ) ∞ n=1 ⊆ {0, 1} is not eventually periodic, then at least one of the numbers ∞ n=1 a n 2 −n and ∞ n=1 a n 3 −n is transcendental. In a different vein, Cassels [Cas], answering a question of Steinhaus about Cantor's middle thirds set C, proved that almost every number in C/2 (with respect to the log 2/ log 3-dimensional Hausdorff measure) is normal to every base which is not a power of 3. More generally, it follows from a famous unresolved conjecture of Furstenberg 1 that almost every real number with respect to any continuous ×3-invariant measure on [0, 1] is normal to every base that is not a power of 3.
Studying the independence between different representations of real numbers remains an active area of research that brings together results and techniques from number theory, ergodic theory, and geometric measure theory. Parallel investigations concerning representations of integers appear to be less developed but are no less natural or interesting. It is the purpose of this paper to advance those investigations by demonstrating various forms of independence between different base representations in the non-negative integers. One of the basic principles that underpin our results in this direction states the following: If r and s are multiplicatively independent positive integers (meaning that the quantity log(r)/ log(s) is irrational) and A and B are subsets of the non-negative integers that are structured with respect to base-r and base-s representations, respectively, then A and B lie in general position.
The following unresolved conjecture of Erdős [Erd] exemplifies this heuristic: for all n 9, it is impossible the express the number 2 n in base 3 using only the digits 0 and 1; see [DW, Lag] for some recent progress. Today, Erdős' conjecture is understood as merely a special case of a much broader conjecture that asserts that any infinite set of natural numbers that has a "simple" description in base r must have a "complex" description in base s (see Question 5.2 in Section 5.2 for more details). A related folklore conjecture in number theory [Inc] posits that {0, 1, 82000} is exactly the set of non-negative integers that can be written in bases 2, 3, 4, and 5 using only the digits 0 and 1. A partial answer to this was given recently by Burrell and Yu [BY], who proved that the set A of non-negative integers that can be written in bases 4 and 5 using only the digits 0 and 1 satisfies A ∩ [0, N ] C ε N ε for all ε > 0. In this paper, we aim to 1) introduce a family of multiplicatively structured fractal subsets of the non-negative integers that naturally arise from digit restrictions, and 2) investigate the transversality, or independence, between members of that family that are structured with respect to multiplicatively independent bases. Our investigation is strongly motivated by the heuristic and conjectures mentioned above and by the recent resolutions of a pair of Furstenberg's conjectures concerning notions of geometric and additive transversality of fractal subsets of the reals. Our results give integer parallels of those advancements in the reals, generalize the aforementioned result of Han and Yu, and make progress toward Erdős' conjecture.
Before recounting the relevant history and stating our main results in full generality, we focus our attention on the special case of restricted digit Cantor sets in the non-negative integers. Although restricted digit Cantor sets comprise only a small subclass of the sets that we consider, most of our results are already novel and interesting for this class. In this sense, the following section serves as a preview of our main results.

Preview of the main results
Let N = {1, 2, 3, . . .} and N 0 = {0, 1, 2, . . .}. An integer base-r restricted digit Cantor set is a set of non-negative integers whose base-r expansion includes only digits from a fixed set D ⊆ {0, 1, . . . , r − 1}, i.e., n i=0 a i r i n ∈ N 0 , a 0 , . . . , a n ∈ D . (1.1) The dimension of such a set A is dim A := log |D|/ log r, in the sense that |A ∩ [0, N )| = N dim A+o(1) . We discuss notions of dimension for more general subsets of the non-negative integers in the next section. While a number of arithmetic properties of integer restricted digit Cantor sets are well studied -divisibility [BS], distribution in arithmetic progressions [EMS, Kon], number of prime factors [KMS], character sums [BCS] -much less appears to be known about the relationship between such sets when they are structured with respect to different bases. Let r and s be multiplicatively independent positive integers (that is, log r/ log s is irrational), and let A, B ⊆ N 0 be base-r and base-s restricted digit Cantor sets, respectively. Under these conditions, our results demonstrate that the sets A and B are transverse both in a geometric / probabilistic sense and in an additive combinatorial sense. More precisely, the sets A and B are • geometrically / probabilistically in general position, in the sense that neither A nor B contains the other and in the sense that the size of A ∩ B is at most what is expected if A and B were independent random sets; • additive combinatorially disjoint, in the sense that the cardinality of the sumset A + B is nearly as large as possible, and hence there are only very few coincidences amongst the sums a + b for a ∈ A and b ∈ B. Our main Theorems A and B address the first point, while Theorem C addresses the second. We move now to formulate corollaries of those theorems that clearly demonstrate these notions of independence.
To describe all of the elements of a non-trivial base-5 restricted digit Cantor set in base 17, all 17 digits are required. The following corollary of Theorem A generalizes this observation by showing that restricted digit Cantor structures with respect to multiplicatively independent bases are mutually incompatible.

Corollary of Theorem
The finer question about the size of the intersection A ∩ B is addressed in Theorem B. For N ∈ N, define A N = A ∩ [0, N ) and B N = B ∩ [0, N ). The sets A N and B N would be probabilistically independent if A N ∩ B N /N = A N B N /N 2 . Examples show that the sets A and B can be disjoint, even in the case that both A and B have a large set of allowed digits, so the inequality for all N large can be understood to demonstrate a type of asymptotic probabilistic transversality between the sets A and B. (As explained in the next section, such an inequality can also be interpreted as A N and B N being geometrically in general position.) Theorem B shows that (1.2) holds up to a factor of N ε ; the precise extent to which (1.2) holds remains open and is addressed briefly in Section 5.2.
If X and Y are finite sets of real numbers, then it is easy to check that Equality holds on the left if and only if X and Y are arithmetic progressions of the same step size. When |X + Y | is near this lower bound, inverse theorems in combinatorial number theory provide additive structural information on the sets X and Y . At the other end of the spectrum, equality holds on the right if and only if none of the sums x + y, with x ∈ X and y ∈ Y , coincide. In this case, the sets X and Y lie in general position from an additive combinatorial point of view.
In this context, the inequality can be understood as demonstrating additive combinatorial transversality between the sets A N and B N . Theorem C shows that (1.3) holds up to a factor of N ε ; the extent to which (1.3) holds is unknown and is discussed briefly in Section 5.1.
Corollary of Theorem C. For all ε > 0 and all sufficiently large N , Theorems A, B, and C are more general than the corollaries above might suggest. Indeed, each result applies not only to restricted digit Cantor sets, but to a wider class of integer fractal sets called multiplicatively invariant sets. Moreover, each set can be replaced by a rounded image of itself under any affine transformation of R. Finally, in Theorem C, the sets A and B can be replaced by arbitrary subsets of A and B, and set cardinality can be replaced with a notion of discrete Hausdorff content. We will introduce multiplicatively invariant sets in Section 1.3 and state our main results precisely there, after providing some historical context and motivation for them in the next section.

History and context
In the language of fractal geometry and dynamical systems, Furstenberg [Fur1,Fur2] established a number of conjectures and results that explore the relationship between multiplicative structures with respect to different bases in the real numbers. The notion of structure particularly relevant to this work is that of multiplicative invariance: a set X ⊆ [0, 1] is ×r-invariant if it is closed and T r X ⊆ X, where T r : [0, 1] → [0, 1] denotes the map T r : x → rx mod 1.
We call a set X ⊆ [0, 1] multiplicatively invariant if it is ×r-invariant for some r 2.
One of Furstenberg's first and most well-known results concerning multiplicatively invariant sets is the following theorem, the measure-theoretic analogue of which is the infamous ×2, ×3 conjecture, a central open problem in ergodic theory. Fur1,Theorem 4.2]). If X ⊆ [0, 1] is simultaneously ×2and ×3-invariant, then either X is finite or X = [0, 1].
The numbers 2 and 3 in Theorem 1.1 can be replaced by any pair of multiplicatively independent positive integers r and s. Following Theorem 1.1, Furstenberg conjectured that if X, Y ⊆ [0, 1] are ×rand ×s-invariant, respectively, then X and Y are transverse in more than one sense, some of which are made precise below. While some of Furstenberg's "transversality conjectures" remain open, two of them were resolved recently by Hochman and Shmerkin [HS], Shmerkin [Shm], and Wu [Wu]. Both of these conjectures are particularly relevant to this work, so we will expound on them further now.
In Euclidean geometry, linear subspaces U, V ⊆ R d are said to be in general position (or By analogy, Furstenberg conjectured 2 that if r and s are multiplicatively independent and X and Y are ×rand ×s-invariant subsets of [0, 1], then In this context, Furstenberg's conjectures in (1.4) and (1.5) say that the multiplicative invariance of the sets X and Y can be leveraged to change the result in Marstrand's theorem from concerning the typical sets λX ∩ (ηY + σ) and λX + ηY to concerning the specific ones X ∩ Y and X + Y . In fact, Furstenberg conjectured that for ×rand ×s-invariant sets X and Y , the inequality in (1.6) and equality in (1.7) hold for all non-zero λ and η and all σ. Hochman and Shmerkin resolved the sumset conjecture by proving a stronger result for multiplicatively invariant measures, and several years later Shmerkin [Shm] and Wu [Wu] independently resolved the intersection conjecture. (These works resolved both conjectures for classes of attractors of iterated function systems, too.) Theorem 1.3 ( [Shm, Wu] and [HS]). Let r and s be multiplicatively independent positive integers, and let X, Y ⊆ [0, 1] be ×rand ×s-invariant sets, respectively. For all λ, η ∈ R\{0} and all σ ∈ R, where dim M denotes the upper Minkowski dimension.
In fact, both the intersection and sumset conjectures follow from Shmerkin's main result in [Shm], which concerns the decay of L q norms of certain self-similar measures of dynamical origin. We derive a number of our main theorems from Shmerkin's work, which we elaborate on further in Section 2.3.
2 The intersection conjecture (1.4) is one of several conjectures stated in [Fur2]. The sumset conjecture (1.5) does not, as far as we are aware, appear by Furstenberg in print, but it was known to have originated with him. 3 Marstrand's slicing and projection theorems originally concerns orthogonal projections of subsets of the plane and intersections with lines. Images of the Cartesian product X × Y under orthogonal projections are, up to affine transformations which preserve dimension, sumsets of the form λX + ηY , while intersections of X × Y with lines are affinely equivalent to sets of the form λX ∩ (ηY + σ). Also note that for sufficiently regular sets X and Y , dimH(X × Y ) = dimH X + dimH Y ; see, for example, [Mat,Corollary 8.11].
In an effort to better understand the role that the multiplicative independence between the bases plays in the sumset theorem, it is natural to ask about the sum of sets that are all structured with respect to the same base r. Taking X ⊆ [0, 1] to be those numbers that can be written in decimal with only the digits 0, 1, and 2, we see that the equality in (1.5) need not hold: log 5 log 10 = dim H (X + X) < 2 dim H X = 2 log 3 log 10 .
Nevertheless, it is a consequence of the following theorem of Lindenstrauss, Meiri, and Peres that the dimension of the iterated sumset X+· · ·+X approaches 1 as the number of summands increases.
This theorem demonstrates that the structure captured by multiplicative invariance sits transversely to the additive structure captured by additive closure: because the sumset X 1 + · · · + X n fills out the entire space (with respect to the Hausdorff dimension), the sets X i are not contained in an additively closed set of dimension less than 1.
While there is a strong historical precedent for the study of ×r-invariant subsets of the unit interval, less seems to be known in the integer and p-adic settings, despite the fact that many of the same objects and questions can be naturally formulated there.
Furstenberg [Fur2], assuming a positive answer to one of his yet-unresolved transversality conjectures in the reals, drew a connection between the real and integer regimes by showing that given any finite word from the alphabet {0, . . . , 9}, the decimal expansion of the number 2 n contains that word provided that n is sufficiently large. This (conditionally) solves an analogue of Erdős' conjecture mentioned earlier.
The folklore conjecture mentioned in the second paragraph in Section 1 is profitably understood in terms of intersections of restricted digit Cantor sets and, as such, evokes the real transversality conjecture of Furstenberg in (1.4). Burrell and Yu's [BY] results toward a resolution of this conjecture rely heavily on Yu's work in [Yu1] on improvements to Shmerkin and Wu's resolution of Furstenberg's intersection conjecture. Drawing on results in [Yu1], Yu [Yu2] also shows that there are few solutions to the equation x + y = z in which the variables come from different integer restricted digit Cantor sets. Using projection theorems and Newhouse's gap lemma, Yu [Yu4] furthermore proves that there are infinitely many sums of powers of five that can be written as sums of powers of three and four.
The first author proved in [Gla,Theorem 1.4] a discrete analogue of Marstrand's projection theorem, building on the work of Lima and Moreira in [LM1]: for all A, B ⊆ Z satisfying a necessary dimension condition 4 and for Lebesgue-a.e. (λ, η) ∈ R 2 , The condition is that the upper mass dimension of A × B is equal to the upper counting dimension of A × B. The upper mass dimension is defined in (1.12), while the upper counting dimension of A × B is equal to lim sup N→∞ max z∈Z 2 log (A × B) ∩ (z + {−N, . . . , N } 2 ) log N .
where the upper mass dimension, dim M , is defined in (1.12) below, ⌊ · ⌋ denotes the floor function, and ⌊λA + ηB⌋ := ⌊λa + ηb⌋ a ∈ A, b ∈ B . It is reasonable to conjecture by analogy that if A and B are restricted digit Cantor sets with respect to multiplicatively independent bases, then (1.10) would hold for all non-zero λ, η ∈ R. We show that this is indeed the case in Theorem C and its generalizations.

Main results
Our primary goals for this article are to introduce the study of multiplicatively invariant subsets of the non-negative integers and to bring transversality results in the integers more in line with those in the reals by giving full-fledged analogues of Theorems 1.1, 1.3, and 1.4. To that end, we begin by introducing an analogue of a ×r-invariant set for the integers.
Let r ∈ N, r 2. Define R r : N 0 → N 0 and L r : N 0 → N 0 by R r : n → ⌊n/r⌋ and L r : n → n − r k ⌊n/r k ⌋, where k = ⌊log n/ log r⌋ when n 1. The maps R r and L r are best understood using the base-r representations of non-negative integers: if n = a k r k + · · · + a 1 r + a 0 , a k = 0, is the base-r representation of n, then R r (n) = a k r k−1 + · · · + a 2 r + a 1 and L r (n) = a k−1 r k−1 + · · · + a 1 r + a 0 .
In other words, the map R r "forgets" the least significant digit (the right-most digit, hence the letter R) while the map L r "forgets" the most significant digit (the left-most digit, hence the letter L) in base r. For example, in base r = 10 we have R 10 (71393) = 7139 and L 10 (71393) = 1393.
It may be helpful to note that a ×r-invariant set A need not satisfy rA ⊆ A and that there are examples showing that the condition rA ⊆ A does not yield a natural integer analogue of the notion of ×r-invariance on the unit interval; see Section 4.4.
There are many natural examples of ×r-invariant subsets of N 0 . Integer base-r restricted digit Cantor sets, defined in (1.1), are clearly ×r-invariant. More general examples arise from symbolic subshifts of {0, 1, . . . , r − 1} N 0 . For any closed and left-shift-invariant set Σ ⊆ {0, 1, . . . , r − 1} N 0 , the corresponding language set is defined by Any language set naturally embeds into the non-negative integers as w 0 r k + · · · + w k−1 r + w k w 0 w 1 · · · w k ∈ L(Σ) , yielding a set that is ×r-invariant. For more details, see Definition 3.9 and Proposition 3.10, and for more such examples, see Examples 3.11. As yet another source of ×r-invariant subsets of the non-negative integers, we note that if X is a ×r-invariant subset of [0, 1], then the set can be shown to be ×r-invariant; see Section 3.4 for more details.
Our first result in the integer setting is an analogue of Theorem 1.1 that demonstrates that there are no non-trivial examples of sets which exhibit structure simultaneously with respect to multiplicatively independent bases. We define [X] δ := {z ∈ R | ∃x ∈ X with |z − x| δ} to be the δ-neighborhood of the set X.
Theorem A. Let r and s be multiplicatively independent positive integers, and let A, B ⊆ N 0 be ×rand ×s-invariant sets, respectively. If λ, η > 0, σ, τ ∈ R and δ > 0 are such that To measure the size of multiplicatively invariant subsets of N 0 and their sumsets and Cartesian products, we make use of two notions of dimension in the integers that parallel the classical Minkowski and Hausdorff dimensions from geometric measure theory. The discrete analogue of the lower and upper Minkowski dimension are the lower and upper mass dimension, defined for A ⊆ N d 0 as where the discrete γ-Hausdorff content, H γ 1 , is defined in Definition 2.2. If these two quantities agree then we say that the discrete Hausdorff dimension of A, dim H A, exists and is equal to this quantity.
The mass dimension and the upper discrete Hausdorff dimension are systematically studied along with a host of other discrete dimensions in [BT2]. We discuss these notions of dimension and the interplay between them at greater length in Section 3.1. For the current discussion, it is helpful to know that and that for any ×r-invariant set A ⊆ N 0 , both the mass dimension dim M A and the discrete Hausdorff dimension dim H A exist and coincide; see Proposition 3.6. Our next main results in the integer setting demonstrate geometric and additive combinatorial transversality between ×rand ×s-invariant subsets of integers. Thus, these results parallel the results of Hochman and Shmerkin, Shmerkin, and Wu by verifying analogues of Furstenberg's intersection and sumset conjectures.
Let r and s be multiplicatively independent positive integers, and let A, B ⊆ N 0 be ×rand ×s-invariant sets, respectively. Define γ = max 0, dim H A + dim H B − 1 .
Theorem B. For all ε, λ, η > 0, σ, τ ∈ R, and sufficiently large N ∈ N, In particular, for all λ, η > 0 and σ, τ ∈ R, The upper bound on the dimension of the set ⌊λA + τ ⌋ ∩ ⌊ηB + σ⌋ in Theorem B provides a satisfactory analogue in the integers to the result of Shmerkin and Wu in (1.8) in the reals. Theorem B will be derived as a corollary of Theorem 4.3, a stronger result in which we demonstrate that the upper bound on ⌊λ A ∩ [0, N ) + τ ⌋ ∩ ⌊η B ∩ [0, N ) + σ⌋ is uniform over a compact set of scaling parameters.
Our next theorem gives an integer analogue of the result of Hochman and Shmerkin in (1.9). We bound both the cardinality and the discrete Hausdorff content of the set ⌊λA ′ +ηB ′ ⌋ from below in terms of the cardinality and the discrete Hausdorff content of the product set Theorem C. For all ε, λ, η > 0, γ ∈ [0, 1], sufficiently large N and non-empty Just as with Theorem B, we derive Theorem C from a more general result, Theorem 4.6, which demonstrates that the inequalities in Theorem C hold uniformly over the scaling parameters λ and η.
Our final main result in the integer setting is an analogue of Theorem 1.4 concerning the dimension of iterated sumsets of ×r-invariant sets.
In the same way as in the continuous regime, this theorem demonstrates that the structure captured by ×r-invariance in N 0 sits transversely to the additive structure captured by additive closure. It also demonstrates the connection between ×r-invariant subsets of the integers and ×r-invariant subsets of [0, 1], and it will serve to emphasize the role multiplicative independence plays in the other results in this section.

Overview of the paper
The paper is organized as follows. In Section 2, we derive the intersection and sumset transversality results for multiplicatively invariant subsets of [0, 1] from the main result in [Shm]. We begin Section 3 with the basic facts and results we need from discrete fractal geometry in Section 3.1 and continue by connecting ×r-invariant subsets of N 0 to symbolic dynamics and multiplicatively invariant subsets of the reals. Section 3 lays the groundwork for Section 4, where we prove our main results: Theorems A, B, C, and D. We construct an example in Section 4.4 that demonstrates that Theorem C is not expected to hold under weaker assumptions. Finally, we conclude the paper with Section 5 by outlining a number of open problems and directions.

Acknowledgements
The authors extend a debt of gratitude to Pablo Shmerkin and Mark Pollicott, whose insightful questions led to improvements in early formulations of Theorem C, and to Sam Chow for help with Section 5.5. The third author is supported by the National Science Foundation under grant number DMS 1901453.

Sums and intersections of multiplicatively invariant subsets of the reals
In this section, we prove that subsets of [0, 1] that are multiplicatively invariant with respect to multiplicatively independent bases are both geometrically and additive combinatorially transverse. The main theorems are derived from the main result of Shmerkin [Shm], but we give particular care on emphasizing the "uniformity" in the parameters. This is the only section in the paper in which we draw on classical continuous fractal geometry, so we begin by establishing the basic terminology and results. The set of real numbers, R, is equipped with the usual Euclidean metric, and, for convenience, all product spaces in the work are endowed with the L 1 (taxicab) metric. The distance between x, y ∈ R d is denoted by |x − y|, and the open ball centered at x with radius δ is denoted B(x, δ). Throughout the paper, a measure refers to a non-negative-valued Radon measure on R d . The total mass of a measure µ is µ := µ(R d ), and its support is denoted supp µ. The push-forward of µ under a map ϕ is denoted ϕµ, so that ϕµ(B) = µ(ϕ −1 B) for all measurable sets B.
Finally, given two positive-valued functions f and g, we write f ≪ a 1 ,...,a k g if there exists a constant c > 0 depending only on the quantities a 1 , . . . , a k for which f (x) cg(x) for all x in the domain common to both f and g. We write f ≍ a 1 ,...,a k g if both f ≪ a 1 ,...,a k g and f ≫ a 1 ,...,a k g.

Fractal geometry of sets and measures in Euclidean space
In this subsection, we give a terse summary of the necessary notation, terminology, and basic results from continuous fractal geometry. The reader interested in learning more will find most of this material in Mattila [Mat,Ch. 4]. Throughout this subsection, ρ and γ are positive real numbers and X ⊆ R d is non-empty.
The lower Minkowski dimension, dim M X, is defined analogously with a limit infimum in place of the limit supremum. If the lower and upper Minkowski dimensions agree, then that value is the Minkowski dimension of X, dim M X. It is easy to check that for all ρ < 1, dim M X = lim sup N →∞ log N (X, ρ −N ) log ρ N and similarly for dim M X.
• The discrete Hausdorff content of X at scale ρ and dimension γ is Note that if X is compact, the index set I in the definitions of H γ ρ (X) and H γ ∞ (X) may be taken to be finite.
Remark 2.3. The discrete Hausdorff content tends to the unlimited Hausdorff content in the limit as the scale tends to zero. More precisely, for X ⊆ R d compact and γ 0, It follows that if lim ρ→0 H γ ρ (X) > 0, then dim H X γ. The proof is straightforward; see [GMR,Lemma 2.4].
Denote by [X] δ the δ-neighborhood of X: The Hausdorff distance between two non-empty, compact sets X, Y ⊆ R d is By the Blaschke selection theorem, the set of all non-empty, compact subsets of R d equipped with the Hausdorff distance is a complete metric space.
as was to be shown.
We proceed with a number of straightforward lemmas that describe how the metric entropy and discrete Hausdorff content behave as functions of the set and the scale.
Since X ′ is ρ-separated, there are at most C = C(a, d) > 0 many points of X ′ in any closed ball of radius (a + 1)ρ. It follows that the map π is at most C-to-1, and hence that |Y ′ | ≫ a,d |X ′ |. It also follows that there are at most C many points of Y ′ in any closed ball of radius ρ. Therefore, the set Y ′ can be thinned to a set Y ′′ ⊆ Y ′ that is ρ-separated and that satisfies |Y ′′ | ≫ a,d |Y ′ |. Combining these observations, To show (2.2), let {B i } i∈I be a collection of open balls that covers Y and where B i has diameter r i ρ and i∈I r γ Proof. Replacing ρ with aρ, we may assume without loss of generality in both statements that 0 < a 1. Since 0 < a 1, we have that N X, ρ N X, aρ . To see the reverse inequality, let X ′ ⊆ X be a maximal (aρ)-separated subset of X. Since the set X ′ intersects any ball of diameter ρ in at most ≪ a,d 1 many points, it may be thinned to an ρ-separated subset X ′′ of X ′ with cardinality |X ′′ | ≫ a,d |X ′ |. Therefore, N (X, aρ) = |X ′ | ≪ a,d |X ′′ | N (X, ρ).
as was to be shown.
Proof. Let X ′ ⊆ X be such that ϕ(X ′ ) is a maximal ρ-separated subset of ϕ(X). Since ϕ has Lipschitz constant a, the points of X ′ are ρ/a-separated. Thus, by Lemma 2.6, verifying the first inequality.
, as was to be shown.
Definition 2.8. The real number γ is a Frostman exponent for a measure µ if there exists a constant c > 0 such that for all balls B ⊆ R d , 3) holds only for balls B of diameter greater than / less than ρ, then γ is a Frostman exponent at scales larger than / smaller than ρ, respectively.
The following lemmas are discrete versions of the well-known mass distribution principle and Frostman's lemma. This pair of results describes a close relationship between the discrete Hausdorff content of a set and the Frostman exponents of measures supported on that set. Lemma 2.9 (cf. [BP,Lemma 1.2.8]). Let c, ρ > 0 and µ be a measure on R d . If for all balls Proof. Let ε > 0, and let {B i } i∈I be a cover of supp µ with balls B i of diameter δ i ρ and The conclusion follows because ε > 0 was arbitrary.
Lemma 2.10. There exists a constant c > 0, depending only on the dimension d ∈ N, for which the following holds. For all non-empty, compact X ⊆ [0, 1] d and all ρ, γ > 0 there exists a measure µ supported on X with µ H γ ρ (X) and with the property that for all balls B of diameter at least ρ, µ(B) c(diam B) γ .
Proof. This requires only a small modification to the proof of Frostman's Lemma found in [BP,Lemma 3.1.1]. By adjusting the constant c, it suffices to prove the lemma for ρ of the form 2 −k . Construct the 2-adic tree corresponding to the set X down to level k, so that vertices of the tree correspond to closed, 2-adic cubes that have non-empty intersection with the set X and two vertices are adjacent in the tree if one of the corresponding cubes contain the other. Assign to an edge between vertices on levels k − 1 and k an edge conductance (or capacity) of 2 −kγ . Associate to each leaf of the tree a point in X.
Instead of defining a measure µ on the space of infinite paths through the tree as is done in [BP], we define µ to be an atomic measure supported on the finite set of points in X that are associated to leaves of the tree. Fix a maximal flow from the root of the tree to the leaves. Define the µ-mass of each leaf to be equal to the value of the fixed maximal flow on the edge adjacent to that leaf.
Every 2-adic cube B with 2 −k diam B 2 −1 and with non-empty intersection with X corresponds to an edge in the tree. By the choice of edge conductance and the fact that the maximal flow is a legal flow, µ(B) (diam B) γ . (Note that this inequality also holds for B = [0, 1] d .) A 2-adic grid cover of X with cells of diameter at least 2 −k corresponds to a cut-set of the tree. By the MaxFlow-MinCut theorem, the measure µ has total mass equal to the minimum cut, which is necessarily greater than H γ 2 −k (X), concluding the proof of the lemma.

Multiplicatively invariant sets and restricted digit Cantor sets
In this short subsection, we record some basic facts about multiplicatively invariant subsets of [0, 1] and the subclass of restricted digit Cantor sets.
• The map T r : Multiplicatively invariant sets behave well in regards to dimension: their Hausdorff and Minkowski dimensions agree, and the dimension of a Cartesian products of such sets is the sum of the dimensions of the factors.
Proof. The first fact is proven in [Fur1,Proposition III.1]. The second follows immediately from [Mat,Corollary 8.11] and the fact that dim H X = dim M X.
Restricted digit Cantor sets are important examples of multiplicatively invariant sets, and the natural Bernoulli measures they support will play an important role in the theorems in this section.
• The base-r restricted digit Cantor set with digits from D ⊆ {0, . . . , r − 1} is the set of those real numbers in [0, 1] expressible in base-r using only digits from D.
• The dimension 5 of the measure µ r,D is dim µ r,D := log |D|/ log r. More generally, the dimension of a product of such measures is the sum of the dimensions of the factors.
The dimensions of a product of restricted digit Cantor sets C r,Dr × C s,Ds and of its associated product measure µ := µ r,Dr × µ s,Ds coincide and are equal to log r/ log |D r | + log s/ log |D s |. In fact, such a measure µ is highly regular, in the sense that for all balls B ⊆ R 2 of diameter 0 < δ < 1 centered at a point in the support of µ, where the asymptotic constants are independent of δ. This follows from the fact that such an estimate holds for single restricted digit Cantor measures, an easy exercise left to the reader. While multiplicatively invariant sets can be vastly more complicated than restricted digit Cantor sets, the following lemma shows that the former can be approximated from above (with respect to dimension) by the latter. The result is well known; for a proof, see [Wu,Prop. 9.3].
Lemma 2.14. Let X ⊆ [0, 1] be multiplicatively invariant. For all ε > 0, there exists a restricted digit Cantor set X ′ containing X such that dim H X ′ < dim H X + ε.

A uniform Frostman exponent projection theorem
For t ∈ R, denote by Π t : R 2 → R the oblique projection Π t (x, y) = x + ty. The goal in this subsection is to prove Theorem 2.15 below, which is a result about Frostman exponents for oblique projections of products of restricted digit Cantor measures. We formulate the results with the objective of calling particular attention to the uniformity over the projection parameter t.
Theorem 2.15. Let µ be the product of two restricted digit Cantor measures whose bases are multiplicatively independent. For all compact I ⊆ R\{0} and all 0 < γ < min(dim µ, 1), Let 2 r < s be multiplicatively independent integers, D r ⊆ {0, . . . , r − 1} and D s ⊆ {0, . . . , s − 1} sets of digits, and C r,Dr ⊆ [0, 1] and C s,Ds ⊆ [0, 1] the base-r and base-s restricted digit Cantor sets with allowed digits D r and D s , respectively. Let µ r,Dr and µ s,Ds the restricted digit Cantor measures on C r,Dr and C s,Ds , respectively, and let µ = µ r,Dr ×µ s,Ds .
We will prove Theorem 2.15 for the measure µ by first proving the following theorem, which we derive from a careful application of Shermkin's recent L q -dimension projection theorem [Shm,Theorem 11.1]. Denote by P m the dyadic partition of R into intervals of length 2 −m , and denote by log the base-2 logarithm.
Theorem 2.16. For all q ∈ (1, ∞) and all compact I ⊆ R\{0}, Proof. It suffices to prove Theorem 2.16 for intervals I ⊆ (0, ∞). Indeed, note that the set 1 − C s,Ds = C s,Ds is a base-s restricted digit Cantor set with digits fromD s = s − 1 − D s whose associated restricted digit Cantor measure µ s,Ds is the image of the measure µ s,Ds under x → 1 − x. It follows that for t < 0, Π t µ is a translate of Π −t (µ r,Dr ⊗ µ s,Ds ), a measure which satisfies the conclusion of the theorem. To prove the theorem for I ⊆ (0, ∞), it suffices to prove it for every interval I of the form I = [ξ, ξs), where ξ > 0, since every compact subset of (0, ∞) is contained in a finite union of intervals of this form. Let ξ > 0 and λ = 1/r. Let T : [0, 1) → [0, 1) be the irrational rotation by β = log r/ log s, T x = x + β mod 1. Let ∆ r and ∆ s be the normalized counting measures on D r and D s , respectively, and for x ∈ [0, 1), define Given x ∈ [0, 1) and n ∈ N, define µ x,0 = δ 0 and µ x,n = µ x,n−1 * λ n ∆(T n x), where we denote by λν the pushforward of the measure ν under the map "multiplication by λ." To each x ∈ X, we associate the measure The tuple ([0, 1), T, ∆, λ) is an example of what Shmerkin calls a "pleasant model." As such, it follows from [Shm,Theorem 11.1] that for all q ∈ (1, ∞), and ν q q = y∈R ν({y}) q . To finish the proof of Theorem 2.16, we will show that for all x ∈ [0, 1) and all q > 1, µ x = Π ξs x µ and α = dim µ. To see that for each x ∈ [0, 1), the measure µ x is equal to Π ξs x µ, observe first that Borrowing notation from Shmerkin, let n ′ (x) := ⌊x + nβ⌋; it is a function of both x and n. Note that n ′ (x) can equivalently be described as the cardinality of the set {i ∈ {1, . . . , n} | {x+ iβ} < β}. Now (2.6) becomes Since convolution is commutative, the fact that the orbit {T x, . . . , T n x} visits [0, β) exactly n ′ (x) times and (2.7) imply that which proves that for every x ∈ [0, 1), µ x = lim n→∞ µ x,n = µ r,Dr * ξs x µ s,Ds = Π ξs x µ, as claimed.
To finish the proof, it remains to show that the value α in (2.5) equals the dimension of µ, which is dim µ = dim µ r,Dr + dim µ s,Ds = log |Dr| log r + log |Ds| log s . Note that for almost every x < β, ∆(x) q q = i∈Dr j∈Ds 1 |Dr||Ds| q = |D r | 1−q |D s | 1−q , and for all x β, as was to be shown.
Though we have not developed the terminology for it, the conclusion in Theorem 2.16 concerns the L q -dimension of the images of µ under oblique projections. The following lemma allows us to derive from Theorem 2.16 a statement concerning Frostman exponents of the projected measures.
Lemma 2.17 ( [Shm,Lemma 1.7]). Let µ be a probability measure on R, q > 1, and γ > 0. If for all m M , Proof. Note that the inequality in (2.8) rearranges to Thus, for all Q ∈ P m , This gives the desired inequality for those intervals that are elements of the partition P m for m M . Any interval of length 2 −(m+1) ρ < 2 −m is covered by at most two elements of the partition P m+1 , giving the result.
We are now in a position to deduce Theorem 2.15 from Theorem 2.16 and Lemma 2.17.

Geometric transversality in the reals
Here we employ Theorem 2.15 to deduce upper bounds on the metric entropy of intersections of multiplicatively invariant sets. The idea in the proof below is borrowed from [Shm,Lemma 1.8].
Theorem 2.18. Let r and s be multiplicatively independent positive integers, and let X, Y ⊆ [0, 1] be ×rand ×s-invariant sets, respectively. Define γ = max(0, dim X + dim Y − 1). For all compact I ⊆ R\{0} and ε > 0, Proof. Let I ⊆ R\{0} be compact and ε > 0. According to Lemma 2.14, we can embed X and Y into restricted digit Cantor sets of slightly higher dimension. Thus, there exists a product of restricted digit Cantor measures µ of dimension dim From Theorem 2.15, we have that there exists ρ 0 > 0 such that for all ρ < ρ 0 , all t ∈ I, and all x ∈ R, Let ρ < ρ 0 , t ∈ I, and x ∈ R. By (2.4) and the fact that ρ 0 is sufficiently small, every ball of radius ρ centered at a point of supp µ has µ-mass greater than ρ dim µ+ε/4 . Therefore, It follows now from the fact that X × Y ⊆ supp µ and Lemma 2.6 that The limit in the conclusion of the theorem follows.
The following corollary is formulated in a way that will make it convenient to apply in the integer setting.
Corollary 2.19. Let r and s be multiplicatively independent positive integers, and let X, Y ⊆ [0, 1] be ×rand ×s-invariant sets, respectively. Define γ = max(0, dim X + dim Y − 1). For all compact I ⊆ R\{0} and all ε > 0, In particular, for all non-zero λ, η ∈ R, and all σ, τ ∈ R, Proof. Let I ⊆ R\{0} be compact and ε > 0. Denote by π 1 : (x, y) → x the first coordinate projection. The following facts are straightforward to verify: Using these facts in order, we see that there exist c 1 , c 2 > 1 depending only on I such that (2.9) We have need for four more easily-verified facts: . Applying N ( · , ρ) to both sides of (2.9) and using the preceding facts in order, we have that there exists c 3 > 1 depending only on I such that The conclusion of the corollary now follows from Theorem 2.18 by appealing to Lemma 2.6.

Additive transversality of sumsets in the reals
In this subsection, we use Theorem 2.18 to show that sets which are multiplicatively invariant with respect to multiplicatively independent bases are transverse in an additive combinatorial sense. The core ideas here appear in [Shm,Corollary 7.4], and we develop it in the context of the discrete Hausdorff dimension here.
The following lemma is a metric entropy analogue of the useful fact that if the fibers of a map X → Y between finite sets X and Y are uniformly bounded in cardinality, then the image of the map must be large.
Proof. Let X ′ be a ρ-separated subset of X of maximal cardinality so that |X ′ | = N (X, ρ).
Since ϕ(X ′ ) is covered by N (ϕ(X ′ ), ρ)-many balls of radius 2ρ, the set X ′ is covered by N (ϕ(X ′ ), ρ)-many pre-images of balls of radius 2ρ under ϕ. Thus, there exists x ∈ R k such that It follows that as was to be shown.
The inequality in (2.15) says that as long as δ ρ, each of those balls has ν-measure at most c 1 δ γ+γ+ε . Therefore, we have that for all ρ δ < ρ 0 , all t ∈ I, and all x ∈ R, ν Π −1 t (B(x, δ/2)) c 1 δ γ+γ+ε c 2 δ −(γ+ε) = c 1 c 2 δ γ . (2.17) We aim now to deduce (2.13) from (2.17). Let 0 < ρ < ρ 0 , and let ∪ i B i be a cover of Otherwise, all balls in the cover have diameter less than ρ 0 , and it follows then from (2.17) that In either case, we have that where the second inequality follows from the fact that both quantities in the minimum are at most 1. Since the cover was arbitrary, we conclude the inequality in (2.13).
In the statement of the following corollary, it is useful to recall Lemma 2.12: all of the notions of dimension for X, Y , and X × Y coincide, and dim(X × Y ) = dim X + dim Y .
Let ε > 0, and let γ = dim(X ′ × Y ′ ) − γ − 2ε. It follows from Theorem 2.21 that there exists a small ρ 0 > 0 such that for all 0 < ρ < ρ 0 , Consider the first inequality if dim is the Minkowski dimension and the second inequality if dim is the Hausdorff dimension. Because γ + γ + ε = dim(A ′ × B ′ ) − ε, the limit infimum (if dim is a lower dimension) or limit supremum (if dim is an upper dimension) as N tends to infinity of the right hand side is positive. It follows that The inequality in (2.21) now follow from the fact that ε > 0 was arbitrary, concluding the proof.

Discrete fractal geometry and multiplicatively invariant subsets of the integers
In this section, we introduce the notation and terminology involved in the study of fractal geometry in the positive integers and develop the basic results concerning multiplicatively invariant subsets. To prove the results in this section and the transversality results in the next, we relate ×r-invariant subsets of the integers to symbolic subshifts on r symbols and to ×r-invariant subsets of [0, 1].

Notions of dimension for subsets of integers
To measure the size of subsets of N 0 , we will make use of the (upper and lower) mass dimension and the (upper and lower) discrete Hausdorff dimension, defined below. The upper and lower mass dimensions and the upper Hausdorff dimension are treated systematically in [BT2]; we will state the properties we require from these quantities with the aim of making this presentation self-contained. These dimensions join a bevy of other natural notions of dimension for subsets of the integers, integer lattices, and more general discrete sets; see [Nau1,Nau2,BT1,IRUT,LM1].
The upper mass dimension, dim M A, is defined analogously with a limit supremum in place of the limit infimum. If dim M A = dim M A, then this value is the mass dimension of A, dim M A.
• The lower discrete Hausdorff dimension of A is The upper discrete Hausdorff dimension, dim H A, is defined analogously with a limit supremum in place of the limit infimum. If dim H A = dim H A, then this value is the discrete Hausdorff dimension of A, dim H A.
As the notation suggests, the mass and discrete Hausdorff dimensions are defined in analogy to the Minkowski and Hausdorff dimensions, respectively. The analogy becomes clearer on noting that so that the mass and discrete Hausdorff dimensions are capturing, in some sense, the Minkowski and Hausdorff dimensions of the sequence of sets N → A/N in the unit cube. As a word of caution, note that our terminology does not match exactly with the terminology used in [BT2]. What we call the upper discrete Hausdorff dimension is called dim L in [BT2] (see Lemma 2.3 in that paper), while the discrete Hausdorff dimension defined in that work does not appear in our work. Our choice of terminology is motivated by the connections drawn in our work between the discrete and continuous notions of dimension.
and the analogous statement with dim M in place of dim M and limit supremum in place of limit infimum holds. (V) For all r ∈ N, r 2, and the analogous statement with dim H in place of dim H and limit supremum in place of limit infimum holds.
Note that the sets in Examples 3.4 (ii) below show that the statement in (III) does not hold for the lower mass and lower discrete Hausdorff dimensions.
Proof. The statements in (I) through (IV) follow from straightforward calculations which are left to the reader.
Both of the statements in (V) follow from (3.2) and the fact that for all γ 0 and all r K N r K+1 , Indeed, this shows that the limit infimum (resp. limit supremum) of the sequence N → H γ 1 A ∩ [0, r N ) /r N γ is non-zero if and only if the limit infimum (resp. limit supremum) of the sequence N → H γ 1 A ∩ [0, N ) /N γ is non-zero.

Lemma 3.3. For all
and no other comparisons are possible in general.
Proof. It is immediate from the definitions that dim M A dim M A and dim H A dim H A, and the set in Examples 3.4 (i) below shows that neither of these inequalities are, in general, equalities.
To see that dim H A dim M A and that dim H A dim M A, note that by covering A∩[0, N ) d by |A ∩ [0, N ) d | many balls of diameter 1 it follows that If γ > dim M A (resp. γ > dim M A), then the limit infimum (resp. limit supremum) of the right hand side is zero, implying that γ The set in Examples 3.4 (iii) below shows that neither of these inequalities are, in general, equalities.
To see that no other comparisons are possible, it suffices to show that there can in general be no comparison between dim H and dim M . This is demonstrated by the sets in Examples 3.4 (i) and (iii) below.
The following examples are meant to illustrate the extent to which the mass and discrete Hausdorff dimensions relate for subsets of N 0 . These examples do not feature the type of structures that we are concerned with in this work, so we leave some of the details to the reader.
It is quick to check that the mass dimension of A exists and dim M A = 1. On the other hand, by covering A with the intervals in its definition, it can be shown that the discrete Hausdorff dimension of A exists and dim H A = 0. We conclude this section by proving some basic upper and lower bounds on the dimension of product sets.

Proof. The inequality in (3.3) is immediate from the definition of upper mass dimension. To prove the inequality in (3.4), define
Let ε > 0 and N ∈ N. It follows by Lemma 2.10 that there exists a measure µ i supported on ( Consider the product measure µ = µ 1 × · · · × µ d ; it is supported on the set A and has the property that for all balls B of diameter at least N −1 , µ(B) c d diam(B) γ−dε . It follows by Lemma 2.9 and (3.2) that By the definition of the lower discrete Hausdorff dimension, the limit infimum as N tends to infinity of the right hand side of the previous inequality is positive, whereby dim H A γ − dε.
The conclusion of the lemma follows since ε > 0 was arbitrary.

Dimension regularity of multiplicatively invariant sets
In this section, we prove that the mass and discrete Hausdorff dimensions of a multiplicatively invariant set (cf. Definition 1.5) exist and coincide. This is accomplished by adapting an argument of Furstenberg [Fur1, Prop. III.1] from the continuous setting.
Proposition 3.6. If A ⊆ N 0 is multiplicatively invariant, then In particular, the mass and discrete Hausdorff dimensions of A exist and coincide.
Before the proof, we introduce some notation that will be useful throughout this section and the following ones. Fix r ∈ N, r 2, and denote by Λ r the alphabet {0, . . . , r − 1}. An element w ∈ Λ ℓ r is a word of length |w| = ℓ. The set of all finite words is Λ * r = ∪ ∞ ℓ=0 Λ ℓ r , and the set of all infinite words is Λ N 0 r . The empty word is the sole element of the set Λ 0 r . The concatenation of the word w ∈ Λ ℓ r with the word v ∈ Λ k r is denoted by juxtaposition: the word wv is an element of Λ ℓ+k r . We write w k for the word w concatenated with itself k many times. Finally, we write w = w 0 · · · w ℓ−1 to indicate that the letters of w are w 0 , . . . , w ℓ−1 ∈ Λ r , in that order.
The function ( · ) r : Λ * r → N 0 serves as the primary link between subsets of non-negative integers and words. In the following subsection, we will use ( · ) r to connect ×r-invariant subsets of N 0 with symbolic subshifts. Note that ( · ) r is surjective and is injective when restricted to Λ ℓ r for some ℓ ∈ N 0 . As a final ingredient before the proof of Proposition 3.6, we give an equivalent characterization of the lower discrete Hausdorff dimension, dim H .

Lemma 3.7. For all
Proof. It suffices to show that for all finite X ⊆ N 0 , H γ, * 1 (X) ≍ H γ 1 (X), and then appeal to Lemma 3.2 (V). That H γ, * 1 (X) H γ 1 (X) follows immediately from the definitions. To show that H γ, * 1 (X) ≪ H γ 1 (X), use the fact that any interval in N 0 of length ℓ can be covered by at most two intervals of the form (w0 d ) r + [0, r d ), where d = ⌈log r ℓ⌉.
Proof of Proposition 3.6. Suppose A ⊆ N 0 is ×r-invariant. Let γ > dim H A. We will show that lim sup M →∞ |A ∩ [0, r M )|/r M γ < ∞, from which it follows that dim M A γ. Since γ > dim H A is arbitrary, it will follow that dim M A dim H A. It will follow then from Lemma 3.3 that dim H A = dim H A = dim M A = dim M A, which will conclude the proof of the lemma.
According to Lemma 3.7, there exists N ∈ N and a collection of intervals B i = (w (i) 0 d i ) r + [0, r d i ), i ∈ I, that cover A ∩ [0, r N ) and for which i∈I r (d i −N )γ < 1. By prepending zeros onto each w (i) , we may assume that Since A is L r -invariant, applying L r to n between 0 and |w (i 1 ) |-many times (depending on how many initial zeroes there are in the word w (i 1 ) ) to n, we see that (w ′ ) r ∈ A. Repeating the argument with (w ′ ) r ∈ A, there exists i 2 ∈ I . Repeating further, we see that there exist i 1 , . . . , i k ∈ I such that w = w (i 1 ) · · · w (i k ) v, where v ∈ Λ <N r . Using the factorization of words w ∈ Λ M r for which (w) r ∈ A described in the previous paragraph and recalling that −|w (i) Since the final quantity is finite and independent of M , and since M > N was arbitrary, it follows that lim sup M →∞ |A ∩ [0, r M )|/r M γ < ∞, as was to be shown.
Proof. This follows immediately by combining Lemma 3.5 and Proposition 3.6.

Connections to symbolic dynamics
Throughout this subsection, we use σ to denote the left-shift on Λ N 0 r , which is defined by We endow Λ r with the discrete topology and Λ N 0 r with the product (or Tychonoff) topology. In the language of symbolic dynamics, any closed subset of Λ N 0 r satisfying σ(Σ) ⊆ Σ is called a subshift. The language set associated to a subshift Σ is the set of all the finite words, including the empty word, appearing in the elements of Σ, i.e., L(Σ) = w 0 · · · w ℓ−1 w = w 0 w 1 · · · ∈ Σ, ℓ ∈ N 0 .
The language set of any subshift can be naturally embedded into the integers using the map ( · ) r : Λ * r → N 0 defined in the previous subsection. This gives rise to the following definition. Definition 3.9. The r-language set associated to a subshift Σ ⊆ Λ N 0 r is the set A Σ ⊆ N 0 defined by A Σ = (w) r := w 0 r ℓ−1 + w 1 r ℓ−2 + · · · + w ℓ−2 r + w ℓ−1 w = w 0 · · · w ℓ−1 ∈ L(Σ) , where (w) r = 0 when w is the empty word.
The following proposition uses r-language sets to relate ×r-invariant sets with subshifts of Λ N 0 r . It is a generalization of some of the results in [LM1,Section 3], where subsets of integers arising from shifts of finite type are defined and studied.
Proposition 3.10. The r-language set A Σ ⊆ N 0 corresponding to any non-empty subshift Σ ⊆ Λ N 0 r is a ×r-invariant set, and the discrete mass and Hausdorff dimensions of A Σ are equal to the normalized topological entropy of the symbolic subshift (Σ, σ), i.e., Moreover, for any ×r-invariant set A ⊆ N 0 , there exists a subshift Σ ⊆ Λ N 0 r such that A coincides with A Σ , the r-language set associated to Σ.
Proposition 3.10 shows that r-language sets 1) provide us with a natural way of producing examples of ×r-invariant subsets of the non-negative integers; and 2) allow us to employ tools and techniques from symbolic dynamics to study ×r-invariant sets. Before the proof, we give some examples of ×r-invariant subsets of N 0 arising this way.
• Let F be a finite set of finite words from the alphabet Λ r . The shift of finite type with forbidden words F is the subshift of Λ N 0 r consisting of all infinite words that do not contain a word from F as a sub-word. The r-language sets corresponding to shifts of finite type form a natural class of ×r-invariant subsets of N 0 which contains the class of restricted digit Cantor sets defined in (1.1) as a rather special subclass. Integer sets corresponding to subshifts of finite type were also considered by Lima and Moreira in [LM1].
• The classical golden mean shift is the subshift of {0, 1} N 0 consisting of all binary sequences with no two consecutive 1's. This leads to a natural example of a ×2-invariant set A golden ⊆ N 0 consisting of all integers whose binary digit expansion does not contain two consecutive 1's. Since the topological entropy of the golden mean shift is known the equal log((1 + √ 5)/2) (cf. [LM2,Example 4.1.4]), it follows from Proposition 3.10 that the dimension of A golden equals log((1 + √ 5)/2)/ log 2.
• The even shift is the subshift of {0, 1} N 0 consisting of all binary sequences so that between any two 1's there are an even number of 0's. The corresponding ×2-invariant set A even ⊆ N 0 consists of all integers whose binary digit expansion has an even number of 0's between any two 1's. Since the topological entropy of the golden mean shift coincides with the topological entropy of the even shift (cf. [LM2,Example 4.1.6]), we conclude that A even and A golden have the same dimension.
• The prime gap shift is the subshift of {0, 1} N 0 consisting of all binary sequences such that there is a prime number of 0's between any two 1's. This corresponds to the ×2-invariant set A prime ⊆ N 0 of all those numbers written in binary in which there is a prime number of 0's between any two 1's. For example, the first 17 elements of A prime are: 0, 1, 2,4,8,9,16,17,18,32,34,36,64,65,68,72,73. The entropy of the prime gap shift is approximately 0.30293, (cf. [LM2,Exercise 4.3.7]) which implies that the dimension of A prime is approximately 0.437.
Proof of Proposition 3.10. Let Σ ⊆ Λ N 0 r be a subshift, and let A Σ be the associated r-language set. To show that A Σ is ×r-invariant, note first that 0 ∈ A Σ because the empty word is in L(Σ). Let n ∈ A Σ , n 1. Because Σ is shift-invariant, there exists a word w = w 0 · · · w ℓ−1 ∈ L(Σ) such that w 0 = 0 and (w) r = n. We see that Since L(Σ) is closed under prefixes, R r (n) ∈ A Σ , and since Σ is shift-invariant, L r (n) ∈ A Σ . This shows that A Σ is ×r-invariant.
Next we will show (3.5). Since A Σ is ×r-invariant, it follows from Proposition 3.6 that dim H A Σ = dim M A Σ . Therefore, it suffices to verify that dim M A Σ = h top (Σ, T )/ log r. Let L ℓ (Σ) denote the set of words of length ℓ appearing in the language set L(Σ), i.e., It is well known (see, for instance, [Wal,Theorem 7.13 (i)]) that the topological entropy of (Σ, σ) is given by where the limit as ℓ → ∞ on the right hand side is known to exist. We claim that for all ℓ ∈ N 0 , Indeed, the first inequality follows immediately from the fact that ( · ) r : Λ ℓ r → [0, r ℓ ) is injective. For the second inequality, associate to each n ∈ A Σ ∩ [0, r ℓ ) a word w ∈ L(Σ) such that w 0 = 0 and (w) r = n. Since n < r ℓ , |w| ℓ. The second inequality follows then from the fact that the association just described is bijective.
Using the fact that the limit in (3.6) exists, it is a short exercise to show that lim ℓ→∞ log ∪ ℓ k=0 L k (Σ) /ℓ exists and is equal to h top (Σ, σ). It follows from the inequalities in (3.7) that dim M A Σ = h top (Σ, T )/ log r, verifying the equality in (3.5).
Finally, suppose A ⊆ N 0 is a ×r-invariant set. We will prove that there exists a subshift Σ ⊆ Λ N 0 r for which A Σ = A. Let Σ (ℓ) denote the set of all infinite words w 0 w 1 · · · ∈ Λ N 0 r for which (w 0 · · · w ℓ−1 ) r ∈ A, and define Σ := ℓ∈N Σ (ℓ) . (3.8) Being an intersection of closed sets, Σ is closed. From L r (A) ⊆ A, it follows that σ(Σ (ℓ) ) ⊆ Σ (ℓ) , whereby σ(Σ) ⊆ Σ. This proves that (Σ, σ) is a subshift. From the construction, it is clear that A Σ ⊆ A. On the other hand, if w ∈ Λ ℓ r is such that (w) r ∈ A then, using R r (A) ⊆ A, it follows that the word w0 ∞ ∈ Σ. It follows that (w) r ∈ A Σ , showing that We note that the identification of ×r-invariant subsets of N 0 and subshifts of Λ N 0 r given by Proposition 3.10 is not bijective. The subshift Σ defined in (3.8) can be shown to be the largest such that A Σ = A, but in general there can be uncountably many distinct subshifts Σ ′ such that A Σ ′ = A.
As a corollary to Proposition 3.10 we obtain the following result, which plays an important role in most of our main results.
Corollary 3.12. For any ×r-invariant A ⊆ N 0 , the set Proof. Note that A ′ is the largest subset of A satisfying R r (A ′ ) = L r (A ′ ) = A ′ ; in particular, it is ×r-invariant. Therefore, to prove dim M A ′ = dim M A, it suffices to find any subset A ′′ ⊆ A satisfying R r (A ′′ ) = L r (A ′′ ) = A ′′ and dim M A ′′ = dim M A. Appealing to Proposition 3.6, this would also prove that dim H A ′ = dim M A. If dim M A = 0, then there is nothing to show, so let us proceed under the assumption that dim M A > 0. According to Proposition 3.10, we can find a subshift Σ ⊆ Λ N 0 r such that A coincides with the r-language set A Σ associated to Σ. Let µ be an ergodic σ-invariant Borel probability measure on Σ of maximal entropy (the existence of such a measure follows from, eg. [Wal,Theorem 8.2 + Theorem 8.7 (v)]). Let Σ ′′ denote the support of µ, and observe that (Σ ′′ , σ) is a subshift of (Σ, σ) with h top (Σ, σ) = h top (Σ ′′ , σ). Moreover, since µ is ergodic, almost every point in Σ ′′ has a dense orbit (by Birkhoff's ergodic theorem) and almost every point is recurrent (by Poincaré's recurrence theorem). Therefore there exists a point x ∈ Σ ′′ which visits every non-empty open set in Σ ′′ infinitely often.
Let A ′′ ⊆ N 0 be the r-language set associated to Σ ′′ . Since Σ ′′ ⊆ Σ, we have A ′′ ⊆ A. Also, by Proposition 3.10, Since A ′′ is an r-language set, it is ×r-invariant, so we already have the inclusions To prove the reverse inclusions, let n ∈ A ′′ , and let w ∈ L(Σ ′′ ) be such that n = (w) r . Since the point x visits every open set of Σ ′′ infinitely often, the word w appears in x infinitely often. Since x is an infinite word, there exists a letter u ∈ Λ r such that the word wu appears in x and hence in L(Σ ′′ ). Now (wu) r ∈ A ′′ and R r (wu) r = (w) r = n, showing that A ′′ ⊆ R r (A ′′ ).
Since h top (Σ ′′ , σ) > 0, some non-zero letter must appear in L(Σ ′′ ). Invoking again the fact that the word w appears infinitely often in x, there must exist a non-zero letter v ∈ Λ r and a k ∈ N 0 such that the word v0 k w appears in x and hence belongs to L(Σ ′′ ). Now (v0 k w) r ∈ A ′′ and L r (v0 k w) r = n, showing that A ′′ ⊆ L r (A ′′ ).
Here is another corollary of Proposition 3.10, which may be of independent interest. Corollary 3.13. If A ⊆ N 0 is multiplicatively invariant and dim M A = 1, then A = N 0 .
Proof. Suppose A is ×r-invariant with dim M A = 1. It follows from Proposition 3.6 that dim M A = 1. There exists a subshift Σ ⊆ Λ N 0 r such that A coincides with the r-language set coming from Σ and h top (Σ, σ) = log r. However, the only subshift of Λ N 0 r with full entropy is the full shift. Hence Σ = Λ N 0 r , which implies A = N 0 .

Connections to fractal geometry of the reals
The purpose of this subsection is to establish a connection between ×r-invariant subsets of the non-negative integers and ×r-invariant subsets of [0, 1]. Recall that X ⊆ [0, 1] is called ×r-invariant if it is closed and T r X ⊆ X, where T r : x → rx mod 1.
First, we remark that every ×r-invariant subset of [0, 1] can be "lifted" to a ×r-invariant subset of N 0 . Indeed, if X ⊆ [0, 1] is ×r-invariant, then one can show that the set ⌊r k x⌋ | x ∈ X, k ∈ N 0 is ×r-invariant. We will not make use of this fact, so we leave the details to the interested reader. Of more importance to us is the converse direction, stated in the following proposition. Recall from Section 2.1 the definition of Hausdorff distance. Therefore, for large enough k, X k ⊂ [X ′ m ] r −m . On the other hand it is clear that X ′ k ⊂ X k . Finally, since from Lemma 3.15 we have that Proof of Proposition 3.14. Define A ′ := k∈N 0 R k r (A) and X ′ k := (A ′ ∩ [0, r k ))/r k . In view of Lemma 3.16, the sequence k → X k converges with respect to the Hausdorff metric if and only if the sequence k → X ′ k converges. Since A ′ = R r (A ′ ), it follows from Lemma 3.15 that This implies that k → X ′ k is a Cauchy sequence, and hence it is convergent (recall that by the Blaschke selection theorem, the set of all non-empty, compact subsets of [0, 1] equipped with the Hausdorff distance is a complete metric space). Let X ′ = lim k→∞ X ′ k , and note that X ′ ⊆ X.
Next, let us show that X is ×r-invariant. Since L r (A) ⊆ A, a simple computation shows T r (X k ) ⊆ X k−1 . Therefore, using X = lim k→∞ X k , we get that T r (X) ⊆ X.
Finally, we must show dim M X = dim M A. As guaranteed by Corollary 3.12, dim M A = dim M A ′ . By combining part (I) of Lemma 3.15 with Lemma 2.5, we see that where the equality follows from the fact that dim M A = lim k→∞ 1 k log r log N (X k , r −k ) (cf. equation (3.1)). On the other hand, using part (II) of Lemma 3.15, Lemma 2.5, and the fact that dim M A ′ = lim k→∞ 1 k log r log N (X ′ k , r −k ), we see (3.11) Combining (3.10) and (3.11) with the fact that X ′ ⊆ X, we see Since dim M A = dim M A ′ and X ′ ⊆ X, we conclude that dim M X exists and is equal to dim M A.

Transversality between multiplicatively invariant subsets of the integers
In this section, we prove our main results, Theorems A, B, C, and D. We also give an example that shows that part of the conclusion of Theorem 4.6 does not hold under weaker assumptions. As in the other sections, the positive integers r and s are fixed, and the implicit constants appearing in asymptotic notation may depend on r and s without further indication.

Sets which are simultaneously multiplicatively invariant
In this subsection we give a proof of Theorem A. We follow the notation and terminology established in Section 3.2. We say that a non-negative integer n begins with the word w in base s if there exists d ∈ N 0 and n 0 ∈ [0, s d ) such that n = (w) s s d + n 0 . (4.1) If w = w 0 · · · w ℓ−1 and w 0 = 0, this means that the ℓ most significant digits in the base-s expansion of n are w 0 , w 1 , . . . , w ℓ−1 , in order.
Lemma 4.1. For all w ∈ Λ ℓ s , there is an arc I w ⊆ [0, 1) modulo 1 (meaning that I w is an interval when 0 and 1 are identified) with the property that for all x (w) s , the integer ⌊x⌋ begins with w in base s if and only if {log x/ log s} ∈ I w .
Proof. Let w ∈ Λ ℓ+1 s . It follows from (4.1) that a positive integer n begins with w in base s if and only if there exists d ∈ N 0 such that (w) s s d n < (w) s + 1)s d .
Therefore, a positive real number x has the property that ⌊x⌋ begins with w in base s if and only if there exists d ∈ N 0 such that The previous inequality is equivalent to log(w) s log s + d log x log s < log (w) s + 1 log s + d. (4.2) Let I w be the modulo 1 arc from the fractional part of log(w) s / log s to the fractional part of log (w) s + 1 / log s in the positive direction. We see that for all x (w) s , the integer ⌊x⌋ begins with w in base s if and only if (4.2) holds, which happens if and only if log x/ log s ∈ I w .
Recall from Section 2.1 that [A] δ denotes the δ-neighborhood of A.
Lemma 4.2. Let r and s be multiplicatively independent positive integers, and let A ⊆ N 0 be ×r invariant and infinite. If λ, δ > 0, τ ∈ R, and B ⊆ N 0 are such that λA + τ ⊆ [B] δ , then for all w ∈ Λ * s , there exists an integer in B that begins with w in base s.
Proof. Let w ∈ Λ * s , and let I w be the arc from Lemma 4.1. Let I ′ w be the middle 3rd subinterval of I w , and let ξ be the length of I ′ w . Define α = log r/ log s. Since α is irrational, there exists K ∈ N such that the set {iα} | i ∈ {0, . . . , K} is ξ-dense in [0, 1).
Since A is infinite, there exists n ∈ A sufficiently large (to be specified momentarily) and such that λn/s K + τ (w) s + δ + λ. Since A is R r -invariant, n, ⌊n/r⌋, . . . , ⌊n/r K ⌋ are all elements of A. Let i ∈ {0, . . . , K}. Since λA + τ ⊆ [B] δ , the real number λ⌊n/r i ⌋ + τ is within a distance δ of the set B. Therefore, there exists t i ∈ R, |t i | λ + δ, such that λn/r i + τ + t i ∈ B.
By the mean value theorem, ensuring that n is sufficiently large, we see that for all i ∈ {0, . . . , K}, It follows from the fact that log λn/r i / log s = log(λn)/ log s − iα and from our choice of K that there exists i ∈ {0, . . . , K} such that log λn/r i / log s ∈ I ′ w . It follows from (4.3) and the definition of ξ that log λn/r i + τ + t i / log s ∈ I w . By our choice of n and the fact that i K, we have that λn/r i + τ + t i (w) s . Therefore, Lemma 4.1 gives that λn/r i + τ + t i , an integer in B, begins with the word w in base s, as was to be shown.
Proof of Theorem A. Let r and s be multiplicatively independent positive integers, and let A, B ⊆ N 0 be ×rand ×s-invariant sets, respectively. Suppose λ, η > 0, σ, τ ∈ R and δ > 0 are such that λA + τ ⊆ ηB + σ δ . We need to show that then either A is finite or B = N 0 .
Suppose A is infinite; we will argue that B = N 0 . Since B is ×s-invariant, it suffices to show that for all w ∈ Λ * s , there exists an integer in B that begins with w in base s. Let w ∈ Λ * s . It follows from (1.11) that λ ′ A + τ ′ ⊆ [B] δ ′ , where λ ′ = λ/η, τ ′ = (τ − σ)/η and δ ′ = δ/η. Since A is ×r-invariant and infinite, Lemma 4.2 gives that some integer in B begins with w ∈ Λ * s in base s, as was to be shown.

Intersections of multiplicatively independent invariant sets
In this subsection, we prove Theorem B, showing that ×rand ×s-invariant sets are geometrically transverse in the sense that the dimension of the intersection of one with any affine image of the other is small. In fact we prove the following stronger version. In particular, for all λ, η, σ, τ ∈ R, Proof. Let I ⊆ R\{0} be compact and ε > 0. Since λ A∩ 0, N +τ ⊆ λ A∩ 0, N +τ 1 and η B ∩ 0, N + σ ⊆ η B ∩ 0, N + σ 1 , the cardinality in the numerator on the left hand side of (4.4) is bounded from above by which is quickly seen to be equal to Define for every k, ℓ ∈ N the sets Define k N := ⌊log N/ log r⌋ + 1 and ℓ N := ⌊log N/ log s⌋ + 1, and note that Therefore, the expression in (4.6) is bounded from above by Since I ⊆ R\{0} is compact, there exists t > 1 such that I ⊆ ±[t −1 , t]. If λ and η belong to I, then λr 1−{log N/ log r} and ηs 1−{log N/ log s} belong to J := ±[t −1 , max(r, s)t]. Therefore, to show (4.4), it suffices to prove In view of Proposition 3.14, the limits X := lim k→∞ X k and Y := lim ℓ→∞ Y ℓ exist in the Hausdorff metric. Moreover, X and Y are ×rand ×s-invariant, respectively, and Put a = max J, and note that for all λ, η ∈ J and σ, τ ∈ R, (4.8) We can now manipulate the left hand side of (4.7) using (4.8), Lemma 2.6, and Corollary 2.19 (with J as I), to get lim sup This verifies (4.7) and concludes the proof of (4.4).
Remark 4.4. We note two modifications to the statement of Theorem 4.3 that can be proved with minor corresponding modifications made to the proof. First, the initial interval [0, N ) can be replaced by an interval symmetric about the origin, (−N, N ). Though A and B consist of positive integers, this is meaningful because the theorem allows for λ and/or η to be negative. Second, using the floor function to round to the integer lattice is a mere convenience: the result hold when the sets λX + τ and ηY + σ are rounded to any other discrete subgroup (or translate of a discrete subgroup) of R. This shows that the key behind the results is of a geometric nature rather than an arithmetic one.

Sums of multiplicatively independent invariant sets
In this subsection, we prove Theorem C, showing that sets which are multiplicatively invariant with respect to multiplicatively independent bases are transverse in an additive combinatorial sense. The results can be phrased in terms of the size (cardinality or Hausdorff content) of finite subsets of multiplicatively invariant sets. The upper bounds on the size of the sumsets are contained in Lemma 4.5 and follow from general considerations. The difficulty in the main results is in proving the lower bounds, which are handled in Theorem 4.6 and are derived from their continuous counterparts in Theorem 2.21.
The upper bound in (4.9) follows from the fact that |⌊ϕ(A ′ × B ′ )⌋| |ϕ(A ′ × B ′ )| |A ′ × B ′ |, while the upper bound in (4.10) follows from Lemma 2.5 and Lemma 2.7 via To prove the dimension inequality for A, B ⊆ N 0 , note that there exists M ∈ N, depending only on max(λ, η), such that for all N ∈ N, (4.11) . It follows from (4.9), (4.10), and (4.11) that Considering the first or second inequality (if dim is the discrete Minkowski or Hausdorff dimension, respectively), the limit infimum or limit supremum (if dim is a lower or upper dimension, respectively) of the quantity on the right hand side is equal to zero because γ > dim(A × B). It follows that dim ⌊λA + ηB⌋ γ. This suffices for the conclusion of the lemma since γ > dim(A × B) was arbitrary and since dim ⌊λA + ηB⌋ is clearly bounded from above by 1.
Theorem 4.6. Let r and s be multiplicatively independent positive integers, and let A, B ⊆ N 0 be ×rand ×s-invariant sets, respectively. Define γ = max(0, dim H (A × B) − 1). For all compact I ⊆ (0, ∞), all 0 γ 1, all ε > 0, all sufficiently large N (depending on I, γ, and ε), all non-empty A ′ ⊆ A ∩ [0, N ) and B ′ ⊆ B ∩ [0, N ), and all λ, η ∈ I, and (4.12) Proof. For all k, ℓ ∈ N, define the sets Let X = lim k→∞ X k and Y = lim ℓ→∞ Y ℓ in the Hausdorff metric; Proposition 3.14 gives that these limits exist, that X and Y are ×rand ×s-invariant subsets of [0, 1], respectively, and that dim H X = dim H A and dim H Y = dim H B. For N ∈ N, define k N := ⌊log N/ log r⌋ + 1 and ℓ N := ⌊log N/ log s⌋ + 1, and note that (4.14) By Lemma 3.15, we have that Let I ⊆ (0, ∞) be compact, 0 γ 1, and ε > 0. Define J := [min I, rs max I]. Next we invoke Theorem 2.21 with J in place of I and either ε/2 in place of ε (to prove (4.12)) or ε as it is (to prove (4.13)). Let N be sufficiently large, to be specified later, but in particular so that ρ := 1/N is sufficiently small for Theorem 2.21 to apply (with ε as either ε/2 or ε).
Replacing cardinality and metric entropy with the γ-dimensional discrete Hausdorff content and appealing to (4.18), Lemma 2.5, Theorem 2.21 (with ε as ε), and (4.21) in the same way, we see that This is precisely the inequality in (4.13), completing the proof.
In the following corollary, note that it is a consequence of Corollary 3.8 that all four discrete notions of dimension, dim M , dim M , dim H , dim H , coincide for multiplicatively invariant sets A and B and their Cartesian product A × B. In particular, Corollary 4.7. Let r and s be multiplicatively independent positive integers, and let A, B ⊆ N 0 be ×rand ×s-invariant sets, respectively. For all dim ∈ {dim M , dim M , dim H , dim H } and λ, η ∈ (0, ∞), dim ⌊λA + ηB⌋ = min 1, dim(A × B) . (4.22) Moreover, for all A ′ ⊆ A and B ′ ⊆ B, Proof. First, note that (4.22) is a consequence of (4.23) and (4.24). Indeed, setting A ′ = A and B ′ = B, if dim A + dim B 1 then (4.22) becomes (4.23), and if dim A + dim B > 1 then (4.24) implies that dim ⌊λA + ηB⌋ 1. Since any subset of N 0 has dimension at most 1, (4.22) follows in this case as well.
Define γ = max(0, dim H (A × B) − 1), and let A ′ ⊆ A and B ′ ⊆ B. To show (4.23) and (4.24), it suffices to show Indeed, this is the lower bound in (4.24), and the upper bound guaranteed by Lemma 4.5 combined with this lower bound gives the desired equality in (4.23). Let dim ∈ {dim M , dim M , dim H , dim H } and λ, η ∈ (0, ∞). If dim(A ′ × B ′ ) = 0, the conclusion is immediate, so we can proceed under the assumption that dim(A ′ × B ′ ) > 0.
There exists M ∈ N such that for all N ∈ N, Consider the first inequality if dim is the discrete Minkowski dimension and the second inequality if dim is the discrete Hausdorff dimension. Because γ + γ + ε = dim(A ′ × B ′ ) − ε, the limit infimum (if dim is a lower dimension) or limit supremum (if dim is an upper dimension) as N tends to infinity of the right hand side is positive. It follows that The inequality in (4.25) now follow from the fact that γ = dim(A ′ × B ′ ) − γ − 2ε and ε > 0 was arbitrary, concluding the proof. This example demonstrates that neither R-invariance nor the invariance indicated in (II) suffice to obtain the result in Corollary 4.7. This is in contrast to Theorem A, where the conclusion holds under the weaker assumption that the sets A and B are R r -and R s -invariant, respectively. We do not know whether L-invariance alone suffices in either Theorem A or Corollary 4.7, but invariance under multiplication by r and s (in the sense of (II)) does not suffice to reach the conclusions in either theorem: the set of squares is invariant under multiplication by both 4 and 9 simultaneously, but has dimension equal to 1/2, while the sets A and B above demonstrate that Corollary 4.7 does not hold under the assumption of invariance under multiplication.

An example that shows R-invariance does not suffice
In what follows, the interval notation [a, b] is understood to mean [a, b]∩ N 0 . For i, j ∈ N 0 , let and then define First we will verify (I) by showing that the mass dimension of A exists and is equal to 1/2; the argument for B is the same. It is easy to see that for all N 1, from which it follows that r N/2 A ∩ [0, r N ) (N + 1) 2 (r (N +1)/2 + 1).
This shows that dim M A = dim M A = dim M A = 1/2. It is clear from the definition of the sets A and B that (II) holds. Next we will verify (III) by showing that R r (A) = A; the same argument works to show that R s (B) = B. Since rA ⊆ A, we have that Since 0 ∈ A, we need only to verify that for all i, ℓ 0, R r (r ℓ I i ) ⊆ A. If ℓ 1, then R r (r ℓ I i ) = r ℓ−1 I i ⊆ A. If ℓ = 0 and i = 0, then we see R r (I 0 ) = {0} ⊆ A. If ℓ = 0 and i 1, Finally we will verify (IV) by showing that for all N sufficiently large, (4.27) where the sum is over all i, j, ℓ, m 0 for which i + ℓ N and σ(j + m) N . We will estimate this sum from above by splitting the sum indices into two sets depending on the "type" of the pair (i, j), which we now define.
Using the trivial bound |C + D| |C||D| for finite sets C, D ⊆ N 0 , we see that if i, j, ℓ, and m are such that (i, j) is of Type I, then (4.29) Using the fact that σj N and that N is sufficiently large, we see from (4.29) that (i−1)/2 > N/4. It follows then from the fact that i + ℓ N that Similarly, using that i N and the fact that N is sufficiently large, we see from (4.29) that σ(j − 1)/2 > N/4. It follows from the fact that σ(j + m) N that (4.31) Now we are in a position to use the following fact: if C, D ⊆ N 0 are contained in intervals of length L, M , respectively, then C + D is contained in an interval of length L + M and hence |C + D| L + M + 1. If i, j, ℓ, and m are such that (i, j) is of Type II, then |r ℓ I i + s m J j | r ℓ+(i+1)/2 + s m+(j+1)/2 + 1.
Using (4.30) and (4.31), we have that Finally, by splitting up the sum in (4.27) into tuples for which the pairs (i, j) are of Type I or Type II, we see by combining (4.28) and (4.32) that the desired inequality in (4.26) holds.

Iterated sums of a multiplicatively invariant set
In this section, we will prove Theorem D. The strategy is to use tools from Section 3.4 to derive Theorem D from the theorem of Lindenstrauss-Meiri-Peres, Theorem 1.4. Throughout this section, r 2 is fixed and all of the asymptotic notation may implicitly depend on it.
Remark 4.8. There are some useful remarks to make before the proof. Let X 1 , X 2 , . . . , which implies by Lemma 2.5 that Combining (4.35) with (4.37) and (4.38), we see that Letting N tend to infinity and noting that n, and hence k, are fixed, these inequalities combine with Remark 2.3, Lemma 3.2 (V), (4.33), and the fact that dim H (Y n /r k ) = dim H Y n to prove the equality in (4.34).

Open directions
We collect in this section a number of interesting open questions concerning multiplicatively invariant subsets of the non-negative integers. Though these questions and conjectures are stated for arbitrary ×r-invariant subsets of N 0 , many are already open and interesting for the special case of base-r restricted digit Cantor sets.

Positive density for sumsets of full dimension
In [Hoc,Problem 4.10], Hochman asks whether the sumset X + Y of a ×rand a ×sinvariant subset of [0, 1] satisfying dim H X + dim H Y > 1 has positive Lebesgue measure. We remark that a projection theorem of Marstrand [Mar, Theorem I] implies that λX + ηY has positive Lebesgue measure for a.e. (λ, η) ∈ R 2 , suggesting a possible affirmative answer. In [Gla,Theorem 1.4], a version of Marstrand's projection theorem for subsets of the integers was obtained, with Lebesgue measure replaced by the notion of upper natural density. 6 It therefore makes sense to consider the following integer analogue of Hochman's question.
Question 5.1. Let r, s ∈ N be multiplicatively independent, and let A, B ⊆ N 0 be ×rand ×s-invariant, respectively. If dim M A+dim M B > 1, then does the sumset A+B have positive upper natural density?

Small intersections
While Question 5.1 considers the sum A + B when sum of the dimensions is larger than 1, it is also natural to ask about the intersection A ∩ B when the sum of the dimensions is below 1. In this direction, Furstenberg conjectured in [Fur2] that if r, s ∈ N are multiplicatively independent and X, Y ⊆ [0, 1] are ×rand ×s-invariant, respectively, then Furstenberg showed that if this conjecture is true, then any large enough power of 2 contains every digit (in base 10), which is a variant of the conjecture of Erdős [Erd] mentioned in the introduction.
The following question is inspired by Furstenberg's conjecture.
Question 5.2. Let r, s ∈ N be multiplicatively independent, and let A, B ⊆ N 0 be ×rand ×s-invariant, respectively. Is it true that dim A + dim B < 1 =⇒ A ∩ B is finite?
A special case of this question is formulated in [Yu3,Conjecture 6.2]. If the answer to Question 5.2 is positive, then Erdős' conjecture holds (this can be seen by taking r = 2, s = 3, A to be the powers of 2, and B to be a restricted digit Cantor set). A weaker version of this statement was established by Lagarias [Lag].
One can formulate a natural quantitative strengthening of Question 5.2 as follows. Given n, r, k ∈ N, let d r,k (n) be the number of subwords of (n) r of length at most k. Then the answer to Question 5.2 is positive if one can show that lim sup k→∞ lim inf n→∞ log d r,k (n) k log r + log d s,k (n) k log s = 1. (5.1) In fact, it suffices to prove that the expression in (5.1) is greater than or equal to 1. Indeed, by considering n to be a power of r, for any k ∈ N, lim inf n→∞ log d r,k (n)/k log r = log 2k/k log r, whereby the expression in (5.1) is at most 1. We believe the limit in (5.1) as k tends to infinity exists, but this would not be necessary to imply a positive answer to Question 5.2.

Difference sets
For closed subsets X, Y ⊆ [0, 1], working with the difference set X − Y is no harder than working with the sumset X + Y . In particular, proving that in Eq. (1.9) requires no additional work. The story changes in the setting of the non-negative integers, where difference sets are much more cumbersome to handle, ultimately because the fibers of the map (a, b) → a − b are not compact. This observation explains why our main results in the integer setting only deal with sumsets λA + ηB with λ and η both positive, and it naturally leads us to the following question. There are many natural variants and extensions of Question 5.3: one can replace A − B with a more general expression ⌊λA + ηB⌋ for any non-zero real numbers λ, η, or one can replace dim M with dim H . One can ask about combinations of the form ⌊λA ′ + ηB ′ ⌋ for arbitrary subsets A ′ and B ′ of A and B, or one can look only at the positive portion (A−B)∩N of the difference set. Our methods provide an outline for obtaining lower bounds, but upper bounds seem to require a new strategy.

Analogous results for other notions of discrete dimension
The upper Banach dimension (or upper counting dimension, cf. [LM1] and [Gla]) of a set A ⊆ N 0 is dim * A := lim sup In general, we only have the inequality dim * A dim M A, but if A ⊆ N 0 is ×r-invariant, then it can be shown that dim M A = dim H A = dim * A.
Question 5.4. Let r and s be multiplicatively independent positive integers, and let A, B ⊆ N 0 be ×rand ×s-invariant, respectively. Is it true that dim * (A + B) = min dim * A + dim * B, 1 , and / or dim * (A ∩ B) max dim * A + dim * B − 1, 0 .
Note that the lower bound dim * (A + B) min dim * A + dim * B, 1 follows from Theorem C using the fact that dim * dim M .
There are several other ways to define natural notions of dimensions for subsets of N 0 . Barlow and Taylor [BT2] define, for example, a discrete notion of packing dimension. The main results in this article suggest possible analogues for their discrete packing dimension.

Polynomial functions of multiplicatively invariant sets
The dimension of the sumset of affine images of multiplicatively invariant sets A and B is described in Theorem C. It is natural to ask about the extent to which the results in that theorem might hold for the sumset of images of A and B under other functions.
In this subsection, for n ∈ N, denote by A (n) the set of n th -powers of elements of A: A (n) := {a n | a ∈ A}. The following question is a (special case of a) natural polynomial extension of Theorem C.
Question 5.5. Let n, m ∈ N, let r, s ∈ N be multiplicatively independent, and let A, B ⊆ N 0 be ×rand ×s-invariant, respectively. Is it true that dim M A (n) + B (m) = min 1 n dim M A + 1 m dim M B , 1 ? (5.2) When A = B = N 0 , an affirmative answer to Question 5.5 follows from basic facts in number theory. It is easy to see that for any A ⊆ N 0 for which the Minkowski dimension exists, the set A (n) has dimension dim M A (n) = dim M A/n (however it is not true in general that A (n) is ×r-invariant when A is). Thus, for arbitrary sets A and B that satisfy a natural dimension condition (see footnote 4), it follows from the discrete version of Marstrand's projection theorem in (1.10) that dim M ⌊λA (n) + ηB (m) ⌋ is equal to the right hand side of (5.2) for Lebesgue almost every λ, η > 0.
We cannot rule out the possibility that (5.2) holds when n, m 2 for arbitrary sets A and B for which the Minkowski dimensions exist. When A = B and n = m = 2, equality in (5.2) is an infinitary version of a conjecture attributed to Ruzsa; see [CG,Conjecture 5].

Multiplicatively invariant sets in relation to other arithmetic sets in the integers
In this paper, we are concerned with transversality between ×rand ×s-invariant sets whenever r and s are multiplicatively independent. In principle, it makes sense to inquire about transversality (or independence) between any two sets which are structured in different ways.
To keep the discussion short, we restrict to infinite arithmetic progressions (or congruence classes), the set of perfect squares, and the set of primes.
Question 5.6. Let A ⊆ N 0 be a ×r-invariant set, and let P be an infinite arithmetic progression. Is it true that dim M (A ∩ P ) is either 0 or dim M (A)?
The answer is yes for restricted digit Cantor sets. In fact, it is proved in [EMS] that such sets satisfy "good equidistribution properties" in residue classes.
More generally, one could ask about the sum or the intersection of a ×r-invariant set and the image of an arbitrary polynomial with integer coefficients, for instance the set of perfect squares, S = {n 2 | n ∈ N 0 }. Note that dim M S = 1/2. Note that the first expression in this question is a special case of the equality in Question 5.5. In a similar vein, one can ask about intersections with the set of prime numbers, P. Note that dim M P = 1.
Question 5.8. Let A ⊆ N 0 be a ×r-invariant set. Is is true that dim M (A ∩ P) is either 0 or dim M (A)?
Maynard showed in [May] that the answer to Question 5.8 is positive when A is a restricted digit Cantor set where the number of restricted digits is small enough with respect to the base. In fact, he obtains a Prime Number Theorem in such sets, which is stronger than simply dim M (A ∩ P) = dim M A. Question 5.8 is open for general restricted digit Cantor sets, and may be very difficult in general. The methods in this paper do not appear to shed new light on this line of inquiry.

Transversality of multiplicatively invariant sets in the rs-adics
The rs-adics is a non-Archimedean regime in which it is easy to ask questions analogous to those asked in this work. Furstenberg proved in [Fur2,Theorem 3] an analogue of Theorem 1.1 in the rs-adics.
Following Furstenberg, note that the maps R r and R s , with domains extended to Z, are uniformly continuous with respect to the rs-adic metric on Z, and therefore extend to continuous transformations of the set of rs-adic integers, Z rs . As a compact metric space, there is a natural Hausdorff dimension to measure the size of subsets of Z rs . Let us call a set X ⊆ Z rs ×r-invariant if it is closed and R r X ⊆ X.
Question 5.9. Let r and s be multiplicatively independent positive integers, and let X, Y ⊆ Z rs be ×rand ×s-invariant sets, respectively. Is it true that dim H X + Y = min dim H X + dim H Y, dim H Z rs , and / or The upper bound on dim H X ∩ Y in the previous question was conjectured by Furstenberg in [Fur2,Conjecture 3]. A positive answer to these questions would bring transversality results in the rs-adics in line with those in the real and integer settings.