On Waring's problem: Beyond Freĭman's theorem

Let ki∈N$k_i\in {\mathbb {N}}$ (i⩾1)$(i\geqslant 1)$ satisfy 2⩽k1⩽k2⩽⋯$2\leqslant k_1\leqslant k_2\leqslant \cdots$ . Freĭman's theorem shows that when j∈N$j\in {\mathbb {N}}$ , there exists s=s(j)∈N$s=s(j)\in {\mathbb {N}}$ such that all large integers n$n$ are represented in the form n=x1kj+x2kj+1+⋯+xskj+s−1$n=x_1^{k_j}+x_2^{k_{j+1}}+\cdots +x_s^{k_{j+s-1}}$ , with xi∈N$x_i\in {\mathbb {N}}$ , if and only if ∑ki−1$\sum k_i^{-1}$ diverges. We make this theorem effective by showing that, for each fixed j$j$ , it suffices to impose the condition ∑i=j∞ki−1⩾2logkj+4.71.$$\begin{equation*} \sum _{i=j}^\infty k_i^{-1}\geqslant 2\log k_j +4.71. \end{equation*}$$More is established when the sequence of exponents forms an arithmetic progression. Thus, for example, when k∈N$k\in {\mathbb {N}}$ and s⩾100(k+1)2$s\geqslant 100(k+1)^2$ , all large integers n$n$ are represented in the form n=x1k+x2k+1+⋯+xsk+s−1$n=x_1^k+x_2^{k+1}+\cdots +x_s^{k+s-1}$ , with xi∈N$x_i\in {\mathbb {N}}$ .


Introduction
Recent advances in the smooth number technology associated with Waring's problem (see [3]) make possible an investigation of the cognate problem to which Freȋman's theorem provides a qualitative answer.Consider then natural numbers k i (i 1) satisfying 2 k 1 k 2 . ... We address the problem of determining circumstances in which, given j ∈ N, there exists a natural number s = s(j) such that all large integers n are represented in the form Freȋman's theorem, announced in 1949 (see [6]), asserts that such holds if and only if the infinite series k −1 i diverges.A formal proof of this conclusion was given by Scourfield in 1960 (see [11,Theorem 1]).We now provide an effective version of this conclusion.
Theorem 1.1.Let k i ∈ N (i 1) satisfy 2 k 1 k 2 . ... Suppose that s is a natural number for which Then all sufficiently large natural numbers n are represented in the form Since the hypotheses of Theorem 1.1 impose the condition k 2 k 1 2, one obtains an immediate consequence of this theorem that implies Freȋman's theorem.
Corollary 1.2.Let k i ∈ N (i 1) satisfy 2 k 1 k 2 . .., and suppose that j ∈ N. Then whenever s is a natural number for which all sufficiently large natural numbers n are represented in the form By making better use of sharper Weyl exponents available for smaller exponents, most particularly in the situation in which one or more of the k i are equal to 2, it would not be difficult to reduce the number 4.71 occurring in the lower bound (1.1) of the statement of Corollary 1.2.Back of the envelope computations suggest that a number comfortably below 3.5 should be accessible.For larger values of k 1 , and k 2 large compared to k 1 , on the other hand, the conclusion of Theorem 1.1 has strength reflecting the limits of current technology.Standard heuristics from the circle method, meanwhile, suggest that the conclusion of Corollary 1.2 should remain valid provided only that If one is prepared to accept a local solubility condition, then the assumption of square-root cancellation for the mean values of exponential sums encountered in the application of the circle method would reduce the lower bound 4 here to 2, while the most optimistic heuristics would reduce this number further to 1.We now turn to the special case of this variant of Waring's problem involving mixed powers in which the exponents consist of consecutive terms of an arithmetic progression.Thus, when k and r are non-negative integers with k 2, we consider the representation of large positive integers n in the shape with x i ∈ N (1 i s).We denote by R(k, r) the least number s having the property that all large integers n are represented in the form (1.2).In particular, the important number G(k) familiar to aficionados of Waring's problem is equal to R(k, 0).Moreover, the pioneering work of Roth [10,Theorem 2] shows that R(2, 1) 50, which is to say that all large enough integers n have a representation in the shape Theorem 1.3.Let k and r be natural numbers with k 2.Then, uniformly in k and r one has R(k, r) A(r)(k + 1) r+1 , where A(r) = r −1 25 r (r + 1) r+1 .Meanwhile, when r k one has R(k, r) (6k + 6) 2r .
It would appear that the only previous work concerning this problem of such generality hitherto available in the literature is that due to Scourfield [11,Theorem 2].The latter work shows that when k 12, one has R(k, r) C(r)k 4r+1 (log k) 2r , in which C(r) is a quantity depending at most on r, but apparently growing somewhat more rapidly than exp(8r 2 ).Meanwhile, the early work of Roth [10] showing that R(2, 1) 50 has been improved by a sequence of authors over the past seven decades (see [1,2,4,5,12,13,14,15,16,17]).Most recently, Liu and Zhao [8] have shown that R(2, 1) 13.As r increases in equation (1.2), the number of summands required to apply available technology increases rapidly.Thus, recent work of Kuan, Lesesvre and Xiao [7, Theorem 2] asserts that R(2, 2) 133.
We isolate two cases of the representation problem (1.2) for special attention.First, in the case r = 1, we note that Ford [5, Theorems 2 and 3] has shown that R(3, 1) 72, and that for large values of k one has R(k, 1) ≪ k 2 log k.A corollary of Theorem 1.3 improves the order of magnitude of the latter bound.
Thus, when k 2 and s 100(k+1) 2 , all large integers n possess a representation in the shape The cognate problem in which one seeks representations of large integers n in the shape , is considerably more difficult.Here, by taking r = k in Theorem 1.3 we obtain the following conclusion.
For comparison, the aforementioned work of Scourfield [11] would deliver a much weaker bound of the general shape R(k, k) ≪ exp(ck 2 ) for a suitable c > 0. It is worth remarking, however, that the heuristic arguments noted in the discussion following the statement of Corollary 1.2 suggest that one should have bounds of the shape R(k, 1) ≪ k and R(k, k) ≪ e k .
Our proofs of Theorems 1.1 and 1.3 are based on applications of the Hardy-Littlewood method, and the basic infrastructure associated with this treatment is outlined in §2.Then, in §3 we prepare a novel Weyl-type estimate for exponential sums over smooth numbers.This eases our path in subsequent discussions and will likely be of independent interest.We combine this estimate with an upper bound for mean values of smooth Weyl sums in §4, making use of our recent work [3] concerning Waring's problem.Thereby, we obtain an acceptable upper bound for appropriate sets of minor arcs relevant to Theorem 1.1 and the second conclusion of Theorem 1.3.A refinement of this approach in §5 applies for the minor arc contribution needed for the proof of the first conclusion of Theorem 1.3.The corresponding major arc contributions are discussed in §6, the positivity of the singular series requiring some additional discussion in §7.
In this paper the letter p is reserved to denote a prime number.We use the standard notation p h n to indicate that p h |n and p h+1 ∤ n.Also, we write θ for min{|θ − n| : n ∈ Z} and e(z) for e 2πiz .

Preliminary infrastructure
The proofs of Theorems 1.1 and 1.3 make use of the Hardy-Littlewood method, with smooth Weyl sums playing a pivotal role.We denote the set of R-smooth integers not exceeding P by A (P, R), so that We note that the standard theory of smooth numbers shows that whenever η ∈ (0, 1), then there is a positive number c η with the property that card(A (P, R)) ∼ c η P as P → ∞ (see for example [18,Lemma 12.1]).
Fix k i ∈ N (i 1) with 2 k 1 k 2 . ... Let s be a natural number, and define θ = θ s (k) by putting For now, it suffices to remark that we have in mind imposing the condition θ > 2, although we shall later impose more onerous conditions on s.We consider a natural number n sufficiently large in terms of s and k 1 , . . ., k s , and we seek a representation of n in the form When 1 i s, we put and observe that all positive integral solutions of the Diophantine equation (2.2) satisfy the bound x i P i (1 i s).Fix η to be a positive number sufficiently small in terms of s and k 1 , . . ., k s , in a manner that will become clear in due course.Our goal is to establish a lower bound for the number T (n; η) of solutions of the equation (2.2) with x i ∈ A (P i , P η i ) (1 i s).The smooth Weyl sums f i (α) that are key to our arguments are defined by it follows via orthogonality that We derive an asymptotic formula for T (n; η) by means of the circle method, the successful application of which requires the introduction of a Hardy-Littlewood dissection.Write L = log n.We take the set of major arcs K to be the union of the intervals K(q, a) = {α ∈ [0, 1) : |α − a/q| L 1/15 n −1 }, with 0 a q L 1/15 and (a, q) = 1.The set of minor arcs complementary to K is then k = [0, 1) \ K. Our first objective, which we complete in § §4 and 5, is to establish that for a suitable positive number δ, provided that s is suitably large in terms of k, one has an upper bound of the shape (2.7) The major arc asymptotics is then the central theme of § §6 and 7, where we confirm the lower bound again for suitably large values of s, and with n sufficiently large in terms of s, k and η.By combining the bounds (2.7) and (2.8) within (2.6), we conclude that T (n; η) ≫ n θ−1 , so that all large enough integers n possess a representation in the shape (2.2).This confirms the respective conclusions of Theorems 1.1 and 1.3, the only problem remaining being that of determining how large s must be so that the estimates (2.7) and (2.8) hold true.Appropriate bounds on s will be determined in § §4, 5 and 7.

An estimate of Weyl-type
This section concerns estimates for the exponential sums f i (α) of use on sets of minor arcs more general than the arcs k introduced in the previous section.Consider a natural number k 2 and a large positive number P .We take Q to be a parameter with 1 Q P k/2 .The major arcs M(Q) are then defined to be the union of the sets M(q, a; Q) = {α ∈ [0, 1) : |qα − a| QP −k }, with 0 a q Q and (a, q) = 1.The complementary set of minor arcs is then defined by putting m(Q) = [0, 1) \ M(Q).Finally, we make use of the dyadically truncated set of arcs Notice that, as a consequence of Dirichlet's approximation theorem, one has [0, 1) = M(P k/2 ).
Our interest lies in estimates for the exponential sum valid when R is a positive number with R P η for a suitably small positive number η, and α ∈ N(Q).In order to describe these estimates, we recall the concept of an admissible exponent from the theory of smooth Weyl sums.A real number ∆ s is referred to as an admissible exponent (for k) if it has the property that, whenever ε > 0 and η is a positive number sufficiently small in terms of ε, k and s, then whenever 2 R P η and P is sufficiently large, one has Here, the underlying parameter is P and the constant implicit in Vinogradov's notation may depend on ε, η, k and s.One may confirm that for all positive numbers s, there is no loss of generality in supposing that one has max{0, k − s/2} ∆ s k.
In order to facilitate concision, from this point onwards we adopt the extended ε, R notation routinely employed by scholars working with smooth Weyl sums while applying the Hardy-Littlewood method.Thus, whenever a statement involves the letter ε, then it is asserted that the statement holds for any positive real number assigned to ε. Implicit constants stemming from Vinogradov or Landau symbols may depend on ε, as well as ambient parameters implicitly fixed such as k and s.If a statement also involves the letter R, either implicitly or explicitly, then it is asserted that for any ε > 0 there is a number η > 0 such that the statement holds uniformly for 2 R P η .Our arguments will involve only a finite number of statements, and consequently we may pass to the smallest of the numbers η that arise in this way, and then have all estimates in force with the same positive number η.Notice that η may be assumed sufficiently small in terms of k, s and ε.
Associated with a family (∆ s ) s>0 of admissible exponents for k is the number an exponent which satisfies the bound τ (k) 1/(4k).For each positive number s, one then has the related number which we have described elsewhere as an admissible exponent for minor arcs (see the preamble to [3, Theorem 5.2] for a discussion of these exponents).
We recall two consequences of our recent work [3] on Waring's problem.
Lemma 3.2.Suppose that s is a real number with s 4k and ∆ s is an admissible exponent.Then whenever Q is a real number with 1 Q P k/2 , one has the uniform bound Proof.For the sake of concision, write f (α) for f (α; P, R).Suppose first that P 1/(2k) < Q P k/2 .Then the conclusion of [3,Theorem 4.2] shows that whenever s 2, one has and the desired conclusion is immediate.
In order to handle the range of Q with 1 Q P 1/(2k) , we turn to the bounds made available in [20].We take a pedestrian approach sufficient for our subsequent application, though we note that with greater effort the condition s 4k could be relaxed at this point.Suppose first that α ∈ M(Q) for some real number Q satisfying exp((log P ) 1/3 ) Q P 1/(2k) .
When a ∈ Z and q ∈ N satisfy (a, q) = 1 and 0 a q Meanwhile, for α ∈ [0, 1) \ M( 1 2 P k/2 ), we put Υ(α) = 0. Note that when α ∈ N(Q) one has Υ(α) ≪ Q −1 .Then as a consequence of [20, Lemma 7.2], much as in the argument leading to [3, equation (6.3)], we find that when α ∈ N(Q), one has We remark in this context that the constraint k 3 of [3, equation (6.3)] is unnecessary in present circumstances.When instead 1 Q exp((log P ) 1/3 ), we appeal to [20,Lemma 8.5], deducing as in the cognate argument associated with [3, Theorem 6.1] that Again, the constraint k 3 of [3] is unnecessary in present circumstances.Thus, in view of the hypothesis s 4k, it follows that when 1 Q P 1/(2k) one has This again delivers the estimate asserted in the statement of the lemma, since This completes the proof of the lemma.
We obtain a pointwise bound for f (α) = f (α; P, R) when α ∈ m(Q) by application of the Gallagher-Sobolev inequality.Lemma 3.3.Suppose that k 2 and 0 < ρ(k) < 2τ (k)/k.Then, uniformly in 1 Q P k/2 , one has the bound Proof.We consider in the first instance the situation in which P 1/(2k) Q P k/2 .Here, we apply Lemma 3.1 with s = u + tk, where u = 4 k and t is sufficiently large in terms of k.The value of u here has been chosen large enough that the classical theory of Waring's problem is comfortably applicable.With more care one could work with a choice for u little more than k log k.On considering the underlying Diophantine equation, working with the value of u already chosen, it follows from Hua's lemma and a routine application of the circle method along the lines described in [18, Chapter 2] that In particular, the exponent ∆ s−tk = 0 is admissible for k, and thus it follows from (3.2) that ∆ * s = −tkτ (k) is an admissible exponent for minor arcs.We therefore infer from Lemma 3.1 that Consider next a real number α with α ∈ m(Q), and let δ be any real number with |δ| P −k .Suppose, if possible, that α + δ ∈ M(Q/2).In such circumstances, there exist a ∈ Z and q ∈ N with (a, q) = 1, 1 q Q/2 and |q(α + δ) − a| This yields a contradiction, so we are forced to conclude that α + δ ∈ m(Q/2).This observation allows us to estimate f (α) pointwise on m(Q) in terms of mean values for f (α) over m(Q/2).Indeed, as a consequence of the Sobolev-Gallagher inequality (see for example Montgomery [9, Lemma 1.1]), we have Hence, whenever α ∈ m(Q), we infer that where The bound (3.4).Meanwhile, by applying Hölder's inequality, we see that where in which Recall that u = 4 k is even.Then since x∈A (P,R) x k e(βx k ), it follows from (3.8) by considering the underlying Diophantine equations that On recalling (3.3), therefore, we deduce that Meanwhile, since s > u we have v > s, and so it follows from (3.8) via Lemma 3.1 that where On substituting (3.9) and (3.10) into (3.7),we find that On substituting (3.6) and (3.11) into (3.5),we conclude that Thus, whenever α ∈ m(Q), we have f (α) ≪ P Q ε−κ , where We now take t sufficiently large in terms of k and obtain the upper bound This confirms the upper bound that we sought when P 1/(2k) Q P k/2 .In order to handle the range of Q with 1 Q < P 1/(2k) , just as in the proof of Lemma 3.2 we turn to the bounds made available in [20].For the sake of concision we adopt the notation of the proof of the latter lemma.Suppose first that one has (log P ) 60ks Q P 1/(2k) and α ∈ m(Q).Then [20, Lemma 7.2] delivers the bound When instead 1 Q (log P ) 60ks , we appeal to [20,Lemma 8.5], deducing that Thus, in all circumstances, we have the estimate asserted in the statement of the lemma, and the proof is complete.
For the purposes of this paper, we apply a bound for τ (k) sufficient for our applications, though falling very slightly short of the sharpest bound attainable using current technology.In this context, it is useful to introduce the exponent ω = ω(k), defined by When k is equal to 2 or 3, we appeal to the formula (3.1) with the crude bound on admissible exponents available from Hua's lemma (see [18,Lemma 2.5]).Thus, we have the admissible exponent ∆ 2 k = 0 since and hence we deduce via (3.1) that τ (2) 1/8 and τ (3) 3/64.Thus In each of these cases, the desired conclusion again follows from Lemma 3.3.
We finish this section with a formulation of our new minor arc estimate of sufficient flexibility that further applications may be anticipated.Theorem 3.5.Suppose that a ∈ Z and q ∈ N satisfy (a, q) = 1.Then one has f (α; P, R) ≪ P λ −1 + λP −k ω , where λ = q + P k |qα − a|.
Proof.We begin by establishing the superficially weaker assertion that, whenever α ∈ R, c ∈ Z and t ∈ N satisfy (c, t) = 1 and |α − c/t| 1/t 2 , then From this assertion, it follows via a standard transference principle (see for example [23,Lemma 14.1]) that the conclusion of the lemma holds.Suppose then that c and t satisfy the relations (c, t) = 1 and |α − c/t| 1/t 2 .We apply Dirichlet's approximation theorem.Thus, there exist b ∈ Z and r ∈ N with (b, r) = 1 satisfying 1 r P k/2 and |rα − b| P −k/2 .We now put ), and it follows from Lemma 3.4 that When c/t = b/r, it follows from the triangle inequality that . Thus, we have either r 1 2 t or Q 1 2 P k /t.When instead c/t = b/r, we have b = c and r = t, and the same conclusion holds.In either case, therefore, we find that Q = max{r, P k |rα − b|} 1  2 min{t, P k /t}.Thus, we infer from (3.14) that Thus the desired conclusion (3.13) follows, and the proof of the theorem is complete.

The minor arc contribution for ascending powers
We now address the representation problem (2.2) and adopt the notation of §2.In situations wherein k 2 may be substantially larger than k 1 , we apply a Weyl-type estimate only for the exponential sum f 1 (α), estimating the remaining ones in mean.Put ) and note that in view of (2.1), (2.3) and (2.4), one has F 1 (0)G 1 (0) ≍ n θ .We take Q to be a parameter with 1 Q n 1/2 and define a Hardy-Littlewood dissection in accordance with that introduced in §3.Thus, the major arcs M(Q) are defined to be the union of the sets with 0 a q Q and (a, q) = 1, and the associated set of minor arcs are defined by setting m i for each i, these definitions align with those of §3 when considering the smooth Weyl sum f i (α).
We begin by recording a Weyl-type estimate for F 1 (α).
Lemma 4.1.When 1 Q n 1/2 , one has the bound Proof.In view of (4.1), this estimate is immediate from Lemma 3.4.
The mean value estimate that we obtain for G 1 (α) depends on admissible exponent bounds.Here we note that, whenever v is even, the corollary to [21, Theorem 2.1] shows that the exponent ∆ v is admissible for k 4, where ∆ v is the unique positive solution of the equation When k is equal to 2 or 3, the admissible exponents available from Hua's lemma show that the real numbers ∆ v defined via (4.2) are admissible.Of course, much sharper estimates are known in these cases (see [22] for the sharpest available conclusions when k = 3).We note that the exponent ∆ s in [21] corresponds to our ∆ v with v = 2s, owing to the slightly different definition employed therein.We next provide an upper bound for the mean value of |G 1 (α)| over the intermediate set of arcs N(Q).In this context, it is convenient to introduce the quantity , one has the bound where Proof.Define the exponents Then it follows from (4.3) that we have s i=2 and hence an application of Hölder's inequality leads us from (4.1) to the bound where For each index i, the largest even integer not exceeding t i is larger than t i − 2, and hence it follows from (4.2) that there is an exponent ∆ t i admissible for k i with Since N(Q) ⊆ M(Q), we find from (4.6) via Lemma 3.2 that Thus, in view of (2.3) and (4.4), we have where On substituting (4.7) into (4.5),we conclude that where The conclusion of the lemma is therefore immediate from (4.8).
By combining the conclusions of Lemmata 4.1 and 4.2, we obtain a minor arc estimate sufficient for our proof of Theorem 1.1.Here and henceforth, we fix η to be a positive number sufficiently small in terms of k 1 , k 2 , . . ., k s , and ε, in the context of the estimates of this and the previous section relevant for the various admissible exponents encountered.Also, we recall the notation of writing L for log n.
Then there is a positive number δ having the property that Proof.By referring to the definition of k in §2, we see that k ⊂ m(L 1/15 ).When L 1/15 Q n 1/2 , it follows from Lemmata 4.1 and 4.2 that Provided that the hypothesis (4.9) holds, it follows from (4.3) that Then, on recalling (2.5), we may conclude thus far that But k is covered by the sets N(Q) via a dyadic dissection, and we see that We complete this section by addressing the particular situation relevant to the second conclusion of Theorem 1.3.
Then, provided that s (6k + 6) 2r , there is a positive number δ having the property that Proof.We apply Lemma 4.3, observing that by a familiar argument one has Thus, provided that one has then the conclusion of the corollary follows from Lemma 4.3.We observe that the hypothesis r k ensures that and hence e 1/(k+r) < 1 + 1/k.It follows that the lower bound (4.10) is satisfied provided that Thus, on recalling that k r, we conclude that the lower bound (4.10) holds whenever s 2(2eD) r (k + 1) 2r .In view of (3.12), however, a modicum of computation confirms the bound 2 1+1/r eD < 35 for r 2, and hence the upper bound asserted in the corollary holds whenever s (6k + 6) 2r , completing the proof.

An enhanced minor arc estimate
Given exponents k i (i 1) having the property that k i is small for numerous small indices i, one may sharpen the analysis of the minor arcs presented in the previous section.We illustrate the underlying strategy in this section with a consideration of the situation in which with r small.We now put ( In accord with the discussion of the previous section, we have F 2 (0)G 2 (0) ≍ n θ .In all other respects, we adopt the notation of the previous section, with an analysis so similar that we are able to economise on detail.
Lemma 5.1.When 1 Q n 1/2 , one has the bound Proof.As a consequence of Lemma 3.4, it follows from (5.1) that sup in which we put However, by applying a familiar lower bound, we find that .
The conclusion of the lemma follows by substituting this estimate into (5.2).
Proof.Define the exponents Then by following the argument of the proof of Lemma 4.2 mutatis mutandis, we obtain the upper bound where we now write δ i = e 1−Φ 2 +2/k i .Thus we infer that where Ω 2 = e 1−Φ 2 +2/(k(r+1)) = Θ 2 , and this delivers the conclusion of the lemma.
We now combine the conclusions of Lemmata 5.1 and 5.2 much as in the proof of Lemma 4.3.
On recalling (2.5) and (2.1), we find that (5.6) yields the bound A comparison with the concluding part of the argument of the proof of Lemma 4.3, using a dyadic dissection of k into subsets of the shape N(Q), therefore leads us from here to the conclusion of the lemma.

The major arc contribution
The goal of this section is to make progress on establishing the lower bound (2.8) for the contribution of the major arcs to T (n; η).Throughout this section and the next, we work under the hypothesis that The hypotheses available to us in Theorem 1.1 ensure that θ > 3, thereby confirming (6.1) with room to spare.In the first conclusion of Theorem 1.3, meanwhile, we have in particular s > 25 r (k + 1), and thus and the hypothesis (6.1) again holds.On the other hand, in the second conclusion of Theorem 1.3 one has r k and s (6k + 6) 2r , whence a similar argument yields θ > 1 r log ((6k + 6) r ) = log(6k + 6) log 18 > 2, and (6.1) holds once again.The upshot of this discussion is that we are cleared in all circumstances to work henceforth under the assumption that (6.1) holds.Suppose next that α ∈ K(q, a) ⊆ K.The standard theory of smooth Weyl sums (see [19,Lemma 5.4]) shows that there is a positive number c = c(η) such that for 1 i s, one has Also, write S(n; X) = 1 q X q −s U n (q), where U n (q) = q a=1 (a,q)=1 S 1 (q, a)S 2 (q, a) • • • S s (q, a)e(−na/q).(6.3) Then since K has measure O(L 1/5 n −1 ), we see that K F (α)e(−nα) dα = c s J(n; L 1/15 )S(n; L 1/15 ) + O(P 1 P 2 • • • P s n −1 L −1/20 ).(6.4)We show in the next section that the singular series q −s U n (q).(6.5) converges absolutely and uniformly for n ∈ N, and moreover that S(n) ≫ 1 whenever s 4k 1 and the condition (6.1) holds.Moreover, under the latter condition we show further that there is a positive number δ such that Next, on making use of the bound supplied by [18, Lemma 2.8], one finds that Hence, working under the hypothesis (6.1), we deduce from (6.2) that there is a positive number δ such that where A familiar approach paralleling that of [18,Theorem 2.3] shows that Thus, on combining (6.4) with (6.6), (6.7) and (6.8), we conclude that there is a positive number δ for which Subject to our verification in the next section that the lower bound S(n) ≫ 1 holds uniformly in n, we conclude from (6.9) that the lower bound (2.8) holds.In combination with the minor arc estimate (2.7), available from Lemma 4.3 under the hypotheses of Theorem 1.1, we conclude that This completes the proof of Theorem 1.1.In order to establish Theorem 1.3, we observe on the one hand that the upper bound (2.7) follows from Lemma 5.3 when s A(r)(k + 1) r+1 .Also, when r k and s (6k + 6) 2r , the upper bound (2.7) follows from Corollary 4.4.Thus, in either case, we find as before that (6.10) follows in these respective situations, and thus the proof of Theorem 1.3 is now complete.

The singular series
In this section we estimate the singular series, confirming (6.6) and the bounds 1 ≪ S(n) ≪ 1.Our argument parallels the analogous treatment of [11], though we introduce refinements en route.We continue working under the hypothesis (6.1) throughout.
Next, by following the argument underlying the proof of [18, Lemma 2.11], we see that U n (q) is a multiplicative function of q.In view of (6.5), we may rewrite S(n) in the form S(n) = p χ p (n), where the product is over all prime numbers p, and By orthogonality, this Euler factor is related to the number M n (p ν ) of incongruent solutions of the congruence A model for the necessary argument, which is standard, may be found in the discussion associated with [18, Lemma 2.12].The limit (7.2) is seen to exist via the relation (7.1).In particular, the quantity χ p (n) is a non-negative number satisfying the relation χ p (n) = 1 + O(p −1−δ ).We summarise our deliberations thus far in the form of a lemma.
Lemma 7.1.Suppose that (6.1) holds.Then the series (6.5) converges absolutely, and there exists a natural number C with the property that for all integers n, one has We have yet to obtain a lower bound for χ p (n) when p C, a matter to which we now attend.Put D = (k 1 , k 2 , . . ., k s ), the greatest common divisor of k 1 , k 2 , . . ., k s .Define the non-negative integer λ by means of the relation p λ D. Then we have p λ |k i for 1 i s, and there exists an index j with 1 j s for which p λ k j .We show that for each integer n, there is a solution of the congruence x k 1 1 + x k 2 2 + . . .+ x ks s ≡ n (mod p λ+τ ), (7.3) with τ = 1 for odd p, and with τ = 2 for p = 2, in each case with (x j , p) = 1.
In order to establish this last assertion, suppose temporarily that there is an integer n having the property that (7.3) has no solution with (x j , p) = 1.It then follows that the range of the left hand side of (7.3) modulo p λ+τ , with (x j , p) = 1, has at most p λ+τ − 1 elements.In the first instance we assume that p is odd.Then, the theory of power residues shows that the monomial x k j takes (p − 1)/(p − 1, k j ) values modulo p λ+1 as x varies over 1 x p λ+1 with (x, p) = 1.Furthermore, for any index i we see that y k i takes at least 1 + (p − 1)/(p − 1, k i ) values modulo p λ+1 as y varies over 1 y p λ+1 .We now repeatedly apply the Cauchy-Davenport theorem (see [18,Lemma 2.14]), beginning with the values of x k j j , and then adding in the remaining powers step-by-step.On recalling (6.1), we find that with p ∤ x j , the range of the left hand side of (7.3), modulo p λ+1 , contains a number of elements which is at least This yields a contradiction, since 2p λ (p − 1) p λ+1 .Our claim concerning the solubility of the congruence (7.3) is consequently confirmed when p is odd.
We next consider the situation with p = 2, where τ = 2.For some index j with 1 j s, one has 2 λ k j .In (7.3) we take x j = 1.We can solve (7.3) with x i ∈ {0, 1} (1 i s and i = j) provided that s 2 λ+2 .However, we have k 1 2 λ , and hence the condition that s 4k 1 suffices to confirm our claim concerning the solubility of the congruence (7.3) in the case that p = 2.
A routine argument now bounds M n (p ν ) from below.We observe that since p λ k j , a number coprime to p is a k j -th power residue modulo p λ+τ if and only if it is a k j -th power residue modulo p ν , for all ν λ + τ .Let x 1 , x 2 , . . ., x s be a solution of (7.3), with (x j , p) = 1, and let ν be a natural number with ν λ + τ .There are p ν−λ−τ choices for y i with y i ≡ x i (mod p λ+τ ) and 1 y i p ν .For each such choice with 1 i s and i = j, the integer n − s i=1 i =j x k i i is a k j -th power residue modulo p λ+τ , and therefore a k j -th power residue modulo p ν .Thus, we have M n (p ν ) p (s−1)(ν−λ−τ ) , so by (7.2) we see that χ p (n) p −(λ+τ )(s−1) .This lower bound holds for all primes p with p λ D and all n ∈ N provided that s 4k 1 and (6.1) holds.
We summarise these deliberations in the following lemma.
Lemma 7.2.Suppose that s 4k 1 , and (6.1) holds.Then there is a positive number ω having the property that S(n) ω for all n ∈ N.
This lemma completes our analysis of the singular series, and thus we have confirmed all of the properties that were needed to complete the analysis of §6.It is worth noting that the condition s 4k 1 of Lemma 7.2 is automatically satisfied whenever the hypotheses of Theorem 1.1 hold for the exponents k 1 , k 2 , . . ., k s .In order to verify this claim, observe that Thus the hypotheses of Theorem 1.1 can be satisfied only when s > 4k 1 .