On the asymptotic formula in Waring's problem with shifts

We show that for integers $k\geq 4$ and $s\geq k^2+(3k-1)/4$, we have an asymptotic formula for the number of solutions, in positive integers $x_i$, to the inequality $\left|(x_1-\theta_1)^k+\dotsc+(x_s-\theta_s)^k-\tau\right|<\eta$, where $\theta_i\in(0,1)$ with $\theta_1$ irrational, $\eta\in(0,1]$, and $\tau>0$ is sufficiently large. We use Freeman's variant of the Davenport--Heilbronn method, along with a new estimate on the Hardy--Littlewood minor arcs, to obtain this improvement on the original result of Chow.


Introduction
In its classical form, Waring's problem asks whether every positive integer N can be represented as a sum of s kth powers of integers, where s does not depend on N. One generalisation of this problem, studied by Davenport and Heilbronn in the 1940s (see, for example, [8]), was to consider diagonal inequalities of the form where the coefficients are non-zero, not all of the same sign, and not all in rational ratio. In particular, they proved that for a suitable sequence (P n ) ∞ n=1 of large real numbers, with P n → ∞ as n → ∞, the number of integer solutions to (1.1) with 1 ≤ x 1 , . . . , x s ≤ P n is at least cP s−k n , for some constant c > 0. In [9], Freeman developed a version of their method which works for all large values of P . This has become known as Freeman's variant of the Davenport-Heilbronn method, and is now a crucial tool in the study of Diophantine inequalities.
The best previously known bound for this problem is due to Chow, who showed in [6] that the asymptotic formula (1.3) holds for s ≥ 2k 2 − 2k + 3. However, an examination of the arguments underlying Chow's work reveals that the recent proof in [4] of the Main Conjecture in Vinogradov's Mean Value Theorem, by Bourgain, Demeter and Guth, allows this constraint to be improved to s ≥ k 2 + k + 1. Although our method also works for k = 3, it does not improve on the best known value of 11 variables, also due to Chow, in [5].
To prove our result, we approximate the number of solutions to (1.2) by a certain integral over the real line (see Section 2 for details). We use a dissection of the real line into major, minor and trivial arcs, as is usual in the Davenport-Heilbronn method, to evaluate this integral. However, in order to achieve our reduction in the number of variables required, we must also divide our arcs into points with or without good approximations by rationals with small denominators, commonly known as the major and minor arcs in the Hardy-Littlewood method.
The new estimate given in Section 3 extends the method of Wooley in [12] to a setting appropriate to Diophantine inequalities. We first obtain a bound for the contribution to a certain mean value from points without good rational approximations, making use of the aforementioned result of Bourgain, Demeter and Guth. In order to give a more precise statement of our result, we must introduce some notation. As is usual in this area, we use e(z) to denote exp(2πiz). For real numbers P , θ and α, with P large and θ ∈ (0, 1), we define We define v to be the real analogue of the classical Hardy-Littlewood minor arcs: namely, with Q a real parameter satisfying 1 ≤ Q ≤ P , we define v = v Q to be the set {α ∈ R : for a ∈ Z and q ∈ N coprime, |qα − a| ≤ QP −k =⇒ q > Q}. (1.4) Finally, we define the kernel function K(α) = sin(πα) πα 2 , which has the property (see [8,Lemma 4]) that for any real number t, one has R e(tα)K(α)dα = max{0, 1 − |t|}.
We are now in a position to state the following result. Theorem 1.2. For natural numbers s ≥ 2, k ≥ 2, and for θ ∈ (0, 1), we have v |f θ (α)| 2s K(α) dα ≪ P ǫ Q −1 (P s+ 1 2 k(k−1) + P 2s−k ). (1.5) This can be viewed as an analogue of the bound We make use of a variant of Theorem 1.2 (see Theorem 3.1, and the subsequent conclusion in Corollary 3.6), which provides a key input to our application of Freeman's variant of the Davenport-Heilbronn method. The number of variables required to achieve this estimate is smaller than that required by Chow to bound the contribution from points on the minor and trivial arcs, and this enables us to make our improvement as stated in Theorem 1.1. On the major arc, we use Chow's result to obtain the main term in the asymptotic formula, while on the remainder of the minor and trivial arcs, we show that the contribution is negligible. In order to do this, we make use of the measure of the set of points with good rational approximations, noting that these points constitute only a small fraction of any given unit interval.
Experts will recognise that there is the potential to apply the key ideas of this paper to related problems, such as that of counting integral solutions to Diophantine inequalities of the shape where the λ i are real numbers, these being essentially a combination of (1.1) and (1.2). We defer such considerations to a future occasion.
We now present a brief outline of the structure of the remainder of this paper. In Section 2, we introduce the preliminary notation required throughout the paper. In Section 3, we present our new estimate for the contribution from the classical Hardy-Littlewood minor arcs, which ultimately allows us to improve on previously known lower bounds for the number of variables required for the asymptotic formula to hold. In Section 4, we show that negligible contributions are obtained from the remainder of the minor and trivial arcs not covered by Corollary 3.6. In Section 5 we present a result of Chow on the major arc, giving the main term in the asymptotic formula for the number of solutions, thus completing the proof of Theorem 1.1.
The author would like to thank Trevor Wooley for his supervision and for suggesting this line of research, and Sam Chow for helpful conversations.

Preliminary notation
We now introduce the conventions and pieces of standard notation which will be used in this paper. When a statement involves ǫ, we mean that the statement holds for any suitably small value of ǫ > 0. We let θ = (θ 1 , . . . , θ s ), and use the vector notation 1 ≤ x ≤ P to mean that 1 ≤ x i ≤ P for all i. Throughout, we assume that τ is sufficiently large in terms of s, k, θ and η.
Let P = τ 1/k , and let N * (τ ) be the number of solutions to (1.2) with 1 ≤ x ≤ P . A solution which does not meet this condition can have at most one of the variables larger than τ 1/k , and in this situation the remaining variables must each be at most some constant multiple of τ (k−1)/k 2 . Thus, since we may assume that s > k 2 − k + 1, it follows that It therefore suffices to prove that We use the Davenport-Heilbronn kernel K(α; η) = η sin(πηα) πηα Consequently, letting provides a weighted count of the number of solutions to (1.2). To be precise, a tuple (x 1 , . . . , x s ) contributes 1 whenever the left-hand side of (1.2) is equal to zero, and 1 − ζ/η whenever the left-hand side of (1.2) is equal to ζ, for some ζ ∈ (0, η).
The following lemma demonstrates the existence of a certain positive function which provides a bound on the values of the exponential sums we are interested in. Lemma 2.1. Let k ≥ 2 be an integer, and let ξ, θ 1 , θ 2 ∈ (0, 1) with θ 1 irrational. Then there exists a positive real-valued function T (P ), for which T (P ) → ∞ as P → ∞, such that Proof. This is a special case of [6, Lemma 2.2].
We divide up the real line into major, minor and trivial arcs, as is usual in the Davenport-Heilbronn method. We fix a real number ξ ∈ (0, 1), and apply Lemma 2.1 to obtain the function T (P ). We then define We can therefore evaluate the integral (2.2) using the dissection The major arc provides the main term in the asymptotic formula for the number of solutions, while the minor and trivial arcs provide negligible contributions which form the error term.
In order to successfully evaluate the contributions from the central major arc (as in [6, Section 3]), we must use a different kernel function related to K(α; η) to reduce the length of the interval which provides a non-negligible contribution. We define and the upper and lower kernel functions These kernel functions are the same as those obtained in [9, Lemma 1] (applied with a = η − δ and b = η for K − (α), and with a = η and b = η + δ for K + (α), along with h = 1 in both cases). Letting U c (t) denote the indicator function of the interval (−c, c), the conclusion of that lemma gives us the bounds Consequently, it suffices to prove that In the approximations which follow, we need to use the moduli of the above kernel functions, and as such it is helpful to note the following decomposition (see [10,Section 2]). We write where are both non-negative. Using (2.1), we also note that 3. An auxiliary estimate In this section, we achieve a bound on the contribution from the traditional Hardy-Littlewood minor arcs, namely those points which are not close to a rational number with small denominator. In doing so, we improve on Chow's result for the number of variables required for the asymptotic formula (1.3) to hold. We follow closely the method of Wooley in [12, Section 2]. Thus, we firstly obtain an estimate for a related mean value, in the case where we have a single shift θ = θ 1 = . . . = θ s . We then use this result, along with Hölder's inequality, to bound the quantity we are interested in, and to generalise to the case in which the shifts need not be the same.
It is convenient to introduce some further notation for use in this section. We define the exponential sums as well as the polynomials We use to denote the integral over [0, 1] t for a suitable value of t, and we define the integral Let J s,k (P, θ) be the number of solutions of the system Using binomial expansions, and the fact that η ≤ 1, we see that this system is equivalent to the system of Diophantine equations Letting J s,k (P ) denote the number of solutions to (3.2) with 1 ≤ x ≤ P , it is therefore the case that J s,k (P, θ) = J s,k (P ). The quantity J s,k (P ) has been widely studied, originally by Vinogradov in the 1930s, leading ultimately to the recent result of Bourgain, Demeter and Guth (see [4, Theorem 1.1]), who prove that for all s ≥ 1 and k ≥ 4. The case k = 3 is due to Wooley in [14]. (1.4). In later applications we will consider Q = (2k) −1 P 1/4 . We are interested in an estimate for the minor arc portion (in the Hardy-Littlewood sense) of the integral I ± s,k (P, θ). For B ⊂ R, we write This allows us to state the key result of this section.
Theorem 3.1. For natural numbers s ≥ 2, k ≥ 2, and for θ ∈ (0, 1), we have Proof. We would like to rewrite the integral of interest in terms of the function G(β, µ), in order to separate out the x k−1 term and estimate it using the rational approximation properties of points in v.
which, by orthogonality, is equal to 1 if σ s,j (x) = h j for all 1 ≤ j ≤ k − 2, and zero otherwise. For any fixed x ∈ [1, P ] 2s , there is precisely one choice of h ∈ Z k−2 which satisfies the above condition, and by the definition of σ s,j we have |σ s,j (x)| ≤ sP j for 1 ≤ j ≤ k − 2. Hence We can therefore rewrite the minor arc integral in the form Hence, using the triangle inequality, and defining we see that Similarly, writing Then, for 1 ≤ y ≤ P , we observe from (3.8) that Substituting this relation into (3.7), we find that we see that If we let then we can write Evaluating the inner integral of I h (γ, y) using orthogonality, we see that and otherwise ∆ x (µ, γ, h, y) = 0.
Substituting this into (3.13) and using Hölder's inequality, we see that For a general function H : R → R, we write Using the Cauchy-Schwarz inequality, and the decomposition (2.7), we obtain From (2.10), we deduce that Υ(K 1 ) contributes whenever |σ s,k (x)| < δ. Recalling that δ = ηL(P ) −1 ≤ η for sufficiently large P , and using the equivalence of systems (3.1) and (3.2), this implies that Similarly, using (2.11), we have We remark that we also have Υ(K) dα ≪ J s,k (2P ), which allows us to establish the simplified claim (1.5) given in the introduction to this paper. Substituting the above estimates into (3.14) and using (2.5), we see that Returning to (3.10), and noting that I as originally defined in (3.5) does not depend on y, we see that By the definition of L(γ), we have and therefore Substituting this into (3.15), we see that and hence, from (3.6), that

Using (3.3), we conclude that
as required.
In particular, we have v |f θ (µ)| 2s |K ± (µ)| dµ ≪ Q −1 P s+ 1 2 k(k−1)+ǫ whenever s ≤ 1 2 k(k + 1), and v |f θ (µ)| 2s |K ± (µ)| dµ ≪ Q −1 P 2s−k+ǫ whenever s ≥ 1 2 k(k + 1). We now wish to use the above result to bound the minor arc contribution for our shifted Waring's problem. From this point onwards, we fix Q = (2k) −1 P 1/4 . In Corollary 3.2, we use the Cauchy-Schwarz inequality and a trivial estimate in order to limit the number of variables needed to achieve the required bound, which ultimately allows us to prove Theorem 1.1 (in Section 5). We then go on to provide a conjectural further improvement (for k = 10 and k ≥ 12) based on an adaptation of a theorem of Bourgain (arising from the results in [4]). Proof. Fix k ≥ 2, and let s 0 = s 0 (k). We first prove (3.16) in the case s = s 0 . Let .
Note that by the definition of s 0 , and since k > 5/3, we have We have a + b = 1, and ak(k + 1) + 2b = s 0 , so, using the notation introduced in (3.4), and suppressing the dependence on k, P and θ, we can apply Hölder's inequality to see that We evaluate the first term using Theorem 3.1 to get For the second term, we use the decomposition (2.7), along with the Cauchy-Schwarz inequality, to obtain Since the number of solutions to the inequality (x − θ) k − (y − θ) k < δ with 1 ≤ x, y ≤ P is O(P ), we use (2.10) and (2.5) Similarly, using (2.11), we have We therefore see that . Hence, with some rearrangement, and using the definitions of a, b and Q, by (3.17).
For s > s 0 , we then use the trivial estimate to obtain We now present a more sophisticated version of the above argument, which follows a similar structure. For j < k a natural number, we define We require an improved version of Hua's lemma. Since [4, Theorem 4.1] applies equally to the case of exponential sums of suitably separated points, such as the set {x − θ : x ∈ N}, as it does to the integer case, it would seem that the following 'shifted' analogue of [3, Theorem 10] should hold. However, the details of such a result do not yet appear in the literature.  Proof. Fix j and k, and let s 1 = s 1 (k, j). We first prove (3.19) in the case s = s 1 . Let Note that by the definition of s 1 we have We have a + b = 1, and ak(k + 1) + bj(j + 1) = s 1 , so, as in Corollary 3.2, we can apply Hölder's inequality to see that We evaluate the first term using Theorem 3.1 to get For the second term, as in Corollary 3.2, we obtain Combining (2.8) and (2.9) with the shifted Hua's lemma, we see that and therefore that I ± j(j+1)/2 (v) ≪ P j 2 +ǫ . Hence, with some rearrangement, and using the definitions of a, b and Q, by (3.20).
Assuming the shifted Hua's lemma, the same result holds whenever s ≥ s 1 (k).
Proof. By Hölder's inequality, and using Corollary 3.2 or Corollary 3.5, as appropriate, we have

The minor and trivial arcs
On the minor and trivial arcs, we first demonstrate the estimate for shifts θ 1 , θ 2 , θ 3 ∈ (0, 1) with θ 1 irrational, and later use Hölder's inequality to obtain the general case. We subdivide our arcs into those points with good rational approximations and those without, in a manner reminiscent of the classical Hardy-Littlewood method, by defining N a,q , and n = (m ∪ t) \ N.
Those points in n are handled using Corollary 3.6, since n = v ∩ (m ∪ t), while those in N are subdivided yet again on the basis of the size of the exponential sum f θ 3 (α). For some real number t (to be chosen later) satisfying 2k(k − 1)t < 1, let 1 + v). Note that for any v ∈ R, we have (4.1) We use this to bound the contribution to the overall integral from B.
Lemma 4.1. Let t be such that 2k(k − 1)t < 1. For u > 1/(2t), and for any v ∈ R, we have Proof. Note that, by assumption, we have ut > 1/2. Therefore, using (4.1), and recalling that Q = (2k We therefore have the following estimate for those B v contained in the minor arcs. Proof. By choosing t so that 2k(k − 1)t is as close as we like to 1, note that we can always find a u such that 1/(2t) < u < k 2 ≤ s − 2. Applying Lemma 4.1 and (2.3), we see that Consequently, we can add in the contribution from the trivial arcs to show that we have the required estimate on B.
Proof. Combining Lemma 4.2 with (2.6) and (4.1), we find that Since 1/2 < t(s − 2) by our choice of t, we conclude that On B, we use a recent result of Baker which improves on an earlier result of Wooley. Firstly, we define α 0 , . . . , α k by and note that α k = α.
Proof. This is the case A = P 1−t of [2,Theorem 4], which in itself is an improvement of [13,Theorem 1.6] in the light of [4]. The direct conclusion is that for small λ, we have 1 ≤ q ≤ P λ+kt and |qα j − a j | ≤ P −j+λ+kt for 1 ≤ j ≤ k; by choosing λ such that 2(k − 1)(λ + t) < 1 and 0 < ζ < 1 − λ − kt, we reach the conclusion given above. It is also possible to extract from the proof of this result in [2] that the greatest common divisor d = (q, a 2 , . . . , a k ) satisfies d ≤ 2k 2 . Restricting to (q, a 1 , . . . , a k ) = 1 can only reduce the values of q and d, so nothing is lost by doing so. We introduce some more notation. For integers q, a 1 , . . . , a k , write S(q, a) = q x=1 e a k x k + . . . + a 1 x q , and for real numbers β 1 , . . . , β k , write I(β) = P 0 e(β k y k + . . . + β 1 y) dy.
We will use Theorem 4.4 in conjunction with the following lemma.
We now combine the minor and trivial arc estimates to deduce the required result for the whole of B.
Combining the above sums, we achieve the stated result for the whole of B.
We now summarise our conclusion for the whole of the Davenport-Heilbronn minor and trivial arcs in another lemma. Assuming the shifted Hua's lemma, the same result holds whenever s ≥ s 1 (k).
Proof. By symmetry, the results of this section hold equally well when θ 3 is replaced by any other θ i with i ≥ 4. Consequently, applying Hölder's inequality, we see that = o(P s−k ), by Lemmata 4.3 and 4.8. By Corollary 3.6, and using the dissection m ∪ t = N ∪ n, we achieve the desired conclusion.

The major arc
On the major arc M = {α ∈ R : |α| < P ξ−k }, we use a result of Chow, noting that it requires only that the number of variables be greater than k. Proof. This is [6, equation (3.28)].
Combining Lemmata 4.9 and 5.1 with (2.4), we obtain the conclusion of Theorem 1.1. Assuming the shifted Hua's lemma, we would achieve the same result whenever s ≥ s 1 (k). In particular, this would provide a further improvement when k = 10 or k ≥ 12.
School of Mathematics, University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, United Kingdom E-mail address: kirsti.biggs@bristol.ac.uk