Hausdorff dimension of plane sections and general intersections

This paper extends some results of Mattila (J. Fractal Geom. 66 (2021) 389–401 and Ann. Acad. Sci. Fenn. A Math. 42 (2017) 611–620), in particular, removing assumptions of positive lower density. We give conditions on a general family Pλ:Rn→Rm,λ∈Λ$P_{\lambda }:\mathbb {R}^n\rightarrow \mathbb {R}^m, \lambda \in \Lambda$ , of orthogonal projections which guarantee that the Hausdorff dimension formula dimA∩Pλ−1{u}=s−m$\dim A\cap P_{\lambda }^{-1}\lbrace u\rbrace =s-m$ holds generically for measurable sets A⊂Rn$A\subset \mathbb {R}^{n}$ with positive and finite s$s$ ‐dimensional Hausdorff measure, s>m$s>m$ . As an application we prove for Borel sets A,B⊂Rn$A,B\subset \mathbb {R}^{n}$ with positive s$s$ ‐ and t-dimensional$t{\text{-dimensional}}$ measures that if s+(n−1)t/n>n$s + (n-1)t/n > n$ , then dimA∩(g(B)+z)⩾s+t−n$\dim A\cap (g(B)+z) \geqslant s+t - n$ for almost all rotations g$g$ and for positively many z∈Rn$z\in \mathbb {R}^{n}$ . We shall also give an application to the estimates of the dimension of the set of exceptional rotations.


introduction
As in [M5], let P λ : R n → R m , λ ∈ Λ, be orthogonal projections, where Λ is a compact metric space.Suppose that λ → P λ x is continuous for every x ∈ R n .Let also ω be a finite non-zero Borel measure on Λ.These assumptions are just to guarantee that the measurability of the various functions appearing later can easily be checked (left to the reader) and that the forthcoming applications of Fubini's theorem are legitimate.
We shall first prove the following two theorems.There and later we shall identify absolutely continuous measures with their Radon-Nikodym derivatives.For the notation, see Section 2.
and for ω almost all λ ∈ Λ, For p = 2 Theorem 1.1 was proved in [M5] and Theorem 1.2 follows by the same argument.For p > 1 the proofs of both theorems have similar strategy but as an essential new ingredient an argument of Harris from [H] is used, see Section 3.
Theorem 1.2 gives a general version of Marstrand's section theorem.For discussion and references for related results, see [M4,Chapter 6].
We shall also give a version of Theorem 1.2, Theorem 3.1, for product sets and measures and use it to prove the following intersection theorem: and for θ n almost all g ∈ O(n), In [M5] this was proved using Theorem 1.1, with p = 2, and assuming that A and B have positive lower densities.Then also the equality holds in (1.7) and (1.8).In general the opposite inequality can fail very badly, see [F].
Theorem 1.3 follows immediately from Theorem 3.1 once the condition (3.3) is verified for the related projections P g , P g (x, y) = x − g(y), x, y ∈ R n , g ∈ O(n), see Section 4. This was already observed in the arXiv version of [M7], but not in the journal.
Previously the same conclusion was obtained in [M1] under the hypothesis s + t > n, s > (n + 1)/2.Notice that this and the assumption s + (n − 1)t/n > n overlap but neither is implied by the other.I believe that that sole condition s + t > n should suffice.This would be optimal.See [M4,Chapter 7] and [M7] for discussions and references for such intersection problems.
We shall also use the present method to improve an estimate from [M3] for the dimension of the set of exceptional g ∈ O(n), see Theorem 4.1.

Preliminaries
We denote by L n the Lebesgue measure in the Euclidean n-space R n , n ≥ 2, and by σ n−1 the surface measure on the unit sphere S n−1 .The closed ball with centre x ∈ R n and radius r > 0 is denoted by B(x, r) or B n (x, r).The s-dimensional Hausdorff measure is H s and the Hausdorff dimension is dim.The orthogonal group of R n is O(n) and its Haar probability measure is θ n .For A ⊂ R n we denote by M(A) the set of non-zero finite Borel measures µ on R n with compact support spt µ ⊂ A. The Fourier transform of µ is defined by The second equality is a consequence of Parseval's formula and the fact that the distributional Fourier transform of the Riesz kernel k s , k s (x) = |x| −s , is a constant multiple of k n−s , see, for example, [M2], Lemma 12.12, or [M4], Theorem 3.10.These books contain most of the background material needed in this paper.
We shall denote by f # µ the push-forward of a measure µ under a map f : . The restriction of µ to a set A is defined by µ A(B) = µ(A ∩ B).The notation ≪ stands for absolute continuity.
The lower and upper s-densities of A ⊂ R n are defined by If H s (A) < ∞, we have by [M2], Theorem 6.2, (with H s normalized as in [M2]), By the notation M N we mean that M ≤ CN for some constant C. The dependence of C should be clear from the context.By c we mean positive constants with obvious dependence on the related parameters.

Dimension of level sets
We shall now prove Theorem 1.2, the proof for Theorem 1.1 is almost the same.The following argument follows very closely that of Harris in [H] for line sections in the first Heisenberg group.
Proof of Theorem 1.2.Note first that using (2.3) our assumptions imply that P λ♯ (H s A) ≪ L m for ω almost all λ ∈ Λ.
For any λ ∈ Λ the inequality dim P −1 λ {u} ∩ A ≤ s − m for L m almost all u ∈ R m follows for example from [M2], Theorem 7.7.This implies dim P −1 λ {P λ x} ∩ A ≤ s − m for H s almost all x ∈ A whenever P λ♯ (H s A) ≪ L m .Hence we only need to prove the opposite inequalities.
Define µ = 10 −s H s A. Due to (2.3) we may assume that µ(B(x, r)) ≤ r s for x ∈ R n , r > 0, by restricting µ to a suitable subset of A with large measure.We may also assume that A is compact, which makes it easier to verify the measurabilities.
Let 0 < t < s − m, 0 < r < 1, and let B j = B(a j , r) ⊂ R n , j = 1, . . ., j 1 , be such that B(0, 1) ⊂ ∪ j B(a j , r/2) and the balls B j , i = 1, . . ., j 1 , have bounded overlap, that is, there is an integer N, depending only on n, such that any point of R n belongs to at most N balls B j , i = 1, . . ., j 1 .Let 0 < δ < r/2 and let µ j be the restriction of µ to B j .Then we have Here, when where M(f ) is the Hardy-Littlewood maximal function of f and the last inequality follows from its L p boundedness.
It follows that For a, x ∈ R n , r > 0, define T a,r (x) = (x − a)/r and let ν j = r −s T a j ,r♯ (µ j ) ∈ M(B(0, 1)).Then one easily checks that ν j (B(x, ρ)) ≤ ρ s for x ∈ R n and ρ > 0.Moreover, (3.1) P λ♯ µ j p p = r m+p(s−m) P λ♯ ν j p p .To check this we may assume that P λ (x, y) = x for x ∈ R m , y ∈ R n−m .By approximation we may assume that µ j is a continuous function.Then P λ♯ µ j (x) = µ j (x, y) dy, T a j ,r♯ µ j (x, y) = r n µ j ((rx, ry) + a j ) and P λ♯ ν j (x) = r −s P λ♯ T a j ,r♯ µ j (x, y) = r n−s µ j ((rx, ry) + a j ) dy, from which (3.1) follows by change of variable.By (1.4), Hence by the bounded overlap, (r Summing over r = 2 −j , j ≥ k, and using the fact that (p − 1)(s − m − t) > 0, yields By the monotone convergence theorem and Fatou's lemma this gives lim sup Hence for ω almost all λ ∈ Λ and µ almost all x ∈ A, This is the same as (3.6) in [M5] and after that the proof is essentially the same as that of Theorem [M5,Theorem 3.1].
For an application to intersections we shall need the following product set version of Theorem 1.2.There P λ : R n × R l → R m , λ ∈ Λ, m < n + l, are orthogonal projections with the same assumptions as before.
and for ω almost all λ ∈ Λ, Proof.The proof is essentially the same as that of Theorem 1.2.I sketch the main steps.We apply the same argument to µ × ν = 10 −s (H s A) × (H t B) in place of µ = 10 −s H s A. We have again µ(B(x, r)) ≤ r s for x ∈ R n , r > 0 and ν(B(y, r)) ≤ r t for y ∈ R l , r > 0.
From this we conclude as before that and finally for ω almost all λ ∈ Λ and µ × ν almost all (x, y) ∈ A × B, Again, the rest of the proof is essentially the same as that of Theorem [M5, Theorem 3.1]; one applies [M5,Lemma 3.2] to µ × ν in place of µ.
For the opposite direction we now have the inequality dim [M2,Theorem 7.7], but we often have dim A × B > s + t.

Intersections
We now apply Theorem 3.1 to the projections P g , P g (x, y) = x − g(y), x, y ∈ R n , g ∈ O(n), to prove Theorem 1.3 on the Hausdorff dimension of intersections.All we need to do is to check the estimate (3.3), after that the proof runs as that of [M5,Theorem 4.1].The qualitative version of (3.3) was given in the proof of [M6,Theorem 4.2].We just have to check that that argument yields the upper bound we need.For convenience, I give essentially the whole short proof.
Then by the results of Du and Zhang, [DZ], Theorem 2.8, The factor ν(B(0, 1)) is not stated in [DZ], but we shall check it below.For n = 2 this estimate was proved by Wolff in [W], see [M4,Theorem 16.1] where this bound is explicitly stated.To apply Theorem 3.1 we need that the implicit constant here is independent of ν as long as ν ∈ M(B n (0, 1)) and ν(B(x, r)) ≤ r t for x ∈ R n , r > 0.
We now check (4.1), cf. the proof [M4,Proposition 16.3].For R > 1, define ν R by f dν R = f (Rx) dνx.Then ν R (x) = ν(Rx).By [DZ,Theorem 2.3] (for the sphere instead of parabola) for every ε > 0, where the implicit constant depends on ε but not on ν.Hence by duality and the Schwartz inequality, By the definition of ν R this is which is the required estimate.
Dimension estimates for sets of exceptional orthogonal transformations were obtained in [M3].The following theorem improves those estimates except when one of the dimensions s and t is at most (n − 1)/2.Theorem 4.1.Let s and t be positive numbers with s+t > n.
We shall apply Theorem 3.1 with Λ = F and P g (x, y) = x − g(y).By the proof of [M6,Theorem 4.3], | P g♯ (µ × ν)(ξ)| 2 dξ dθg < Cµ(B n (0, 1))ν(B l (0, 1)), with C independent of µ and ν.The right hand side is not explicitly stated in [M6], but it can be checked in the same way as in the proof of Theorem 1.3.Then, as in the proof of Theorem 1.3, it follows from Theorem 3.1 and the formula (4.3) that (4.5) holds for θ almost all g ∈ F .This contradiction completes the proof.

Some comments
For the applications to intersections we only used Theorem 1.2 with p = 2, which, essentially, was already proved in [M5].For a similar method Harris [H] also needed the case p < 2; he proved the L 3/2 estimate for vertical projections in the first Heisenberg group and derived from it the almost sure dimension of line sections.
Orponen proved in [O] an L p estimate for radial projections with some p > 1.If the analog of Theorem 1.1 or 1.2 would be true in this setting, it would improve some of the results of [MO].Although most of the arguments for Theorems 1.1 and 1.2 work much more generally, the scaling argument, in particular (3.1), seems to require right kind of scaling properties of the maps.