Improved \ell^p-Boundedness for Integral k-Spherical Maximal Functions
Theresa C. Anderson, Brian Cook, Kevin Hughes, Angel Kumchev
DD ISCRETE A NALYSIS , 2018:10, 18 pp.
Improved (cid:96) p -Boundedness for Integral k -Spherical Maximal Functions Theresa Anderson Brian Cook Kevin Hughes Angel Kumchev
Received 2 August 2017; Published 29 May 2018
Abstract:
We improve the range of (cid:96) p ( Z d )-boundedness of the integral k -spherical maximalfunctions introduced by Magyar. The previously best known bounds for the full k -sphericalmaximal function require the dimension d to grow at least cubically with the degree k . Combiningideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter,and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application,we improve upon bounds in the authors’ previous work [1] on the ergodic Waring–Goldbachproblem, which is the analogous problem of (cid:96) p ( Z d )-boundedness of the k -spherical maximalfunctions whose coordinates are restricted to prime values rather than integer values. Key words and phrases:
Maximal functions, integral averages, surface measures, Fourier transforms, circlemethod, exponential sums.
Our interest lies in proving (cid:96) p ( Z d )-bounds for the integral k -spherical maximal functions when k ≥
3. Thesemaximal functions are defined in terms of their associated averages, which we now describe. Define a positivedefinite function f on R d by f ( x ) = f d , k ( x ) : = | x | k + · · · + | x d | k , and note that when x ∈ R d + , f ( x ) is the diagonal form x k + · · · + x kd . For λ ∈ N , let R ( λ ) denote the number ofintegral solutions to the equation f d , k ( x ) = λ . (1.1)When R ( λ ) >
0, define the normalized arithmetic surface measure σ λ ( x ) : = R ( λ ) { y ∈ Z d : f ( y ) = λ } ( x ) . We are interested in averages given by convolution with these measures: σ λ ∗ f ( x ) = R ( λ ) (cid:88) f ( y ) = λ f ( x − y ) c (cid:13) cb Licensed under a Creative Commons Attribution License (CC-BY) DOI: 10.19086/da.3675 a r X i v : . [ m a t h . C A ] M a y or functions f : Z d → C .We know from the literature on Waring’s problem that as λ → ∞ , one has the asymptotic R ( λ ) ∼ S d , k ( λ ) λ d / k − , (1.2)where S d , k ( λ ) is a convergent product of local densities: S d , k ( λ ) = (cid:89) p ≤∞ µ p ( λ ) . Here µ p ( λ ) with p < ∞ is related to the solubility of (1.1) over the p -adic field Q p , and µ ∞ ( λ ) to solubilityover the reals. It is known that when d is su ffi ciently large in terms of k , one has1 (cid:46) S d , k ( λ ) (cid:46) . (1.3)In particular, these bounds hold for d ≥ k when k ≥ d ≥ k otherwise (seeTheorems 4.3 and 4.6 in Vaughan [17]).Throughout the paper we use the notation f ( x ) (cid:46) g ( x ) or g ( x ) (cid:38) f ( x ) to mean that there exists a constant C > | f ( x ) | ≤ C | g ( x ) | for all su ffi ciently large x ≥
0. The implicit constant C may depend on ‘inessential’or fixed parameters, but will be independent of ‘ x ’; below the implicit constants will often depend on theparameters k , d and p . For instance, (1.3) means that there exists positive constants C and C depending on d and k so that C ≤ S d , k ( λ ) ≤ C .In view of (1.2) and (1.3), we may replace the convolution σ λ ∗ f above by the average A λ f ( x ) : = λ − d / k (cid:88) f ( y ) = λ f ( x − y ) . (1.4)Our k -spherical maximal function is then defined, for x ∈ Z d , as the pointwise supremum of all averages A ∗ f ( x ) : = sup λ ∈ N | A λ f ( x ) | . (1.5)Variants of this maximal function were introduced by Magyar [9] and studied later in [11, 10, 8, 2, 6, 7]. Inparticular, Magyar, Stein and Wainger [11] considered the above maximal function when k = d ≥ (cid:96) p ( Z d ) when p > dd − . This result is sharp except at the endpoint, for whichthe restricted weak-type bound was proved later by Ionescu [8]. To the best of our knowledge, the sharpestresults on the boundedness of A ∗ for degrees k ≥ k ≥
3, define d ( k ) : = k − max ≤ j ≤ k − (cid:26) k j − min(2 j + , j + j ) k − j + (cid:27) . (1.6)Further, set τ k = max (cid:8) − k , ( k − k ) − (cid:9) and define the function δ ( d , k ) by k δ ( d , k ) : = (cid:40) ( d − d ) / ( k + k − d ) if d ( k ) ≤ d ≤ k + k , + ( d − k − k ) τ k if d > k + k . Finally, define p ( d , k ) : = max (cid:26) dd − k , + + δ ( d , k ) (cid:27) . We remark that when d > d ( k ), p ( d , k ) always lies in the range (1 , heorem 1. If k ≥ , d > d ( k ) and p > p ( d , k ) , then the maximal operator A ∗ , defined by (1.5) , is boundedon (cid:96) p ( Z d ) : that is, (cid:107) A ∗ f (cid:107) (cid:96) p ( Z d ) (cid:46) (cid:107) f (cid:107) (cid:96) p ( Z d ) . Let d ∗ ( k ) : = + (cid:98) d ( k ) (cid:99) denote the least dimension in which Theorem 1 establishes that A ∗ is bounded on (cid:96) ( Z d ). We emphasize that for large k , we have d ∗ ( k ) = k − k + O ( k / ), whereas in previous results, suchas the work of the third author [6, 7], one required d > k − k . While our results improve on these previousresults by a factor of the degree k , the conjecture is that the maximal function is bounded on (cid:96) ( Z d ) for d (cid:38) k (see [6] for details), but such a result appears to be way beyond the reach of present methods.It is also instructive to compare d ∗ ( k ) to the known bounds for the function ˜ G ( k ) in the theory of Waring’sproblem (defined as the least value of d for which the asymptotic formula (1.2) holds). It transpires thatthe values of d ∗ ( k ) match the best known upper bounds on ˜ G ( k ) for all but a handful of small values of k ,and even in those cases, we miss the best known bound on ˜ G ( k ) only by a dimension or two. For an easiercomparison, we list the numerical values of d ∗ ( k ), k ≤
10, their respective analogues in earlier work, and thebounds on ˜ G ( k ) in Table 1. k d ∗ ( k ) 10 16 24 35 47 62 79 97 d ∗ ( k ) 13 33 81 181 295 449 649 901˜ G ( k ) ≤ Table 1: Comparison between the values of d ∗ ( k ) in Theorem 1, the corresponding values d ∗ ( k ) in the work of Hughes[6], and the known upper bounds on ˜ G ( k ) in Bourgain [3] and Vaughan [16]. A key ingredient in the proof of Theorem 1 and its predecessors is an approximation formula generalizing(1.2). First introduced in [11] when k =
2, such approximations are obtained for the average’s correspondingFourier multiplier: (cid:99) A λ ( ξ ) = λ − d / k (cid:88) f ( x ) = λ e ( x · ξ ) , (1.7)where ξ ∈ T d and e ( z ) = e π iz .We need to introduce some notation in order to state our approximation formula. Given an integer q ≥ Z q = Z / q Z and Z ∗ q for the group of units; we also write e q ( x ) = e ( x / q ). The d -dimensional Gausssum of degree k is defined as G ( q ; a , b ) = q − d (cid:88) x ∈ Z dq e q ( a f ( x ) + b · x )for a ∈ Z q and b ∈ Z dq . If Σ λ denotes the surface in R d + defined by (1.1) and dS λ ( x ) denotes the inducedLebesgue measure on Σ λ , we define a continuous surface measure d σ λ ( x ) on Σ λ by d σ λ ( x ) : = λ − d / k dS λ ( x ) |∇ f ( x ) | . We note that d σ λ is essentially a probability measure on Σ λ for all λ . We also fix a smooth bump function ψ , which is 1 on (cid:2) − , (cid:3) d and supported in (cid:2) − , (cid:3) d . Finally, we write (cid:101) µ for the R d -Fourier transform of ameasure µ on R d and (cid:98) f for the Z d -Fourier transform (which has domain T d ) of a function f on Z d .3 heorem 2 (Approximation Formula) . If k ≥ , d > d ( k ) and λ ∈ N is su ffi ciently large, then one has (cid:99) A λ ( ξ ) = ∞ (cid:88) q = (cid:88) a ∈ Z ∗ q e q ( − λ a ) (cid:88) w ∈{± } d (cid:88) b ∈ Z d G ( q ; a , w b ) ψ ( q ξ − b ) (cid:103) d σ λ ( w ( ξ − q − b )) + (cid:99) E λ ( ξ ) , where the error terms (cid:99) E λ are the multipliers of convolution operators satisfying the dyadic maximal inequality (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) | E λ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) Λ − δ (1.8) for each Λ ≥ and all su ffi ciently small δ > . Our Approximation Formula takes the same shape as those in [6] and [7], but with an improved errorterm that relies on two recent developments: • the underlying analytic methods were improved in the authors’ previous work [1], • and the recent resolution of the main conjecture about Vinogradov’s mean value integral [19, 4] andrelated refinements of classical mean value estimates [18, 3].The most dramatic improvement follows from our improved analytic methods originating in [1] where weimprove the range of (cid:96) ( Z d ) by a factor of the degree k . In [7], this sort of improvement - which also used therecent resolution of the Vinogradov mean values theorems [18, 3] - was limited to maximal functions oversu ffi ciently sparse sequences. Here, our bounds supersede those for integral k -spherical maximal functionsover sparse sequences in [7] because our treatment of the minor arcs in the error term is more e ffi cient. Thereader may compare Lemmas 3.2 and 3.1 below to Lemmas 2.1 and 2.2 of [7] to determine the e ffi cacy of ourmethod here. Consequently, [18] and [3] allow us to further improve slightly upon a more direct applicationof the Vinogradov mean value theorems from [19] and [4]. One minor drawback is that in our method the (cid:15) -losses in [19, 4] do not allow us to deduce endpoint bounds.As an application, we deduce that the maximal function of the "ergodic Waring–Goldbach problem"introduced in our recent work [1] is bounded on the same range of spaces as above. That maximal function isassociated with averages where, instead of sampling over integer points, we sample over points where allcoordinates are prime. To be precise, let R ∗ ( λ ) denote the number of prime solutions to the equation (1.1)weighted by logarithmic factors: that is, R ∗ ( λ ) : = (cid:88) f ( x ) = λ P d ( x )(log x ) · · · (log x d ) , where P d is the indicator function of vectors x ∈ Z d with all coordinates prime. When R ∗ ( λ ) >
0, define thenormalized arithmetic surface measure ω λ ( x ) : = R ∗ ( λ ) { y ∈ P d : f ( y ) = λ } ( x )(log x ) · · · (log x d )and the respective convolution operators W λ f : = ω λ ∗ f . (1.9)Similarly to (1.2), we know that as λ → ∞ , one has the asymptotic R ∗ ( λ ) = (cid:0) S ∗ d , k ( λ ) + o (1) (cid:1) λ d / k − , S ∗ d , k ( λ ) is a product of local densities similar to S d , k ( λ ) above. Moreover, when d > k and λ is restrictedto a particular arithmetic progression Γ d , k , we have 1 (cid:46) S ∗ d , k ( λ ) (cid:46)
1, and the above estimate turns into a trueasymptotic formula for R ∗ ( λ ) (see the introduction and references in [1]). By Theorem 6 of [1] and Theorem 1above, we immediately obtain the following result. Here, as in [1], d (3) =
13 and d ( k ) = k + k + Theorem 3.
If k ≥ , d ≥ d ( k ) and p > p ( d , k ) , then the maximal function defined byW ∗ f ( x ) : = sup λ ∈ Γ d , k | W λ f ( x ) | , is bounded on (cid:96) p ( Z d ) . As another application one may give analogous improvements of the ergodic theorems obtained in [6], butwe do not consider this here.To establish our theorems, we follow the paradigm in [11] and strengthen the connection to Waring’sproblem as initiated in [7] by using a lemma from [1]; we then use recent work on Waring’s problem to obtainimproved bounds. We remark that [15] and [12] previously connected mean values (Hypothesis K ∗ andVinogradov’s mean value theorems respectively) to discrete fractional integration. In Section 2, we outlinethe proofs of Theorems 1 and 2; we recall some results from [6, 9] and state the key propositions requiredin the proofs. The remaining sections establish the propositions. In Section 3 we deal with the minor arcs;particularly, in Section 3.2, we use the recent work of Bourgain, Demeter and Guth [4] on Vinogradov’s meanvalue theorem and a method of Wooley [18] for estimation of mean values over minor arcs. In Section 4, weestablish the relevant major arc approximations. Finally, in Section 5, we establish the boundedness of themaximal function associated with the main term in the Approximation Formula. Since A ∗ is trivially bounded on (cid:96) ∞ ( Z d ), we may assume through the rest of the paper that p ≤
2. Fix Λ ∈ N and consider the dyadic maximal operator A Λ f : = sup λ ∈ [ Λ / , Λ ) | A λ f | . When λ ≤ Λ , we have A λ f = λ − d / k (cid:90) T ( h Λ ( θ ) ∗ f ) e ( − λθ ) d θ , where h Λ ( θ ) = h Λ ( θ ; x ) : = e ( θf ( x )) [ − N , N ] d ( x ) with N = Λ / k . This representation allows us to decompose A λ into operators of the form A Bλ f = λ − d / k (cid:90) B ( h Λ ( θ ) ∗ f ) e ( − λθ ) d θ , (2.1)for various measurable sets B ⊆ T .Our decomposition of A λ , is inspired by the Hardy–Littlewood circle method. When q ∈ N and 0 ≤ a ≤ q ,we define the major arc M ( a / q ) by M ( a / q ) = (cid:20) aq − kqN k − , aq + kqN k − (cid:21) .
5e then decompose T into sets of major and minor arcs , given by M = M ( Λ ) = (cid:91) q ≤ N (cid:91) a ∈ Z ∗ q M ( a / q ) and m ( Λ ) = T \ M ( Λ ) . Since the major arcs M ( a / q ) are disjoint, this yields a respective decomposition of A λ as A λ = (cid:88) q ≤ N (cid:88) a ∈ Z ∗ q A a / q λ + A mλ , where A a / q λ : = A M ( a / q ) λ .We will use the notations A B ∗ and A B Λ to denote the respective maximal functions obtained from theoperators A Bλ . For example, from (2.1) and the trivial bound for the trigonometric polynomial h Λ ( θ ) ∗ f , weobtain the trivial (cid:96) -bound (cid:13)(cid:13) A B Λ (cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) Λ | B | . (2.2)In Section 3, we analyze the minor arc term and prove the following result. Proposition 2.1.
If k ≥ , d > d ( k ) , and Λ (cid:38) , then (cid:13)(cid:13) A m Λ (cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) Λ − δ (2.3) for all δ ∈ (0 , δ ( d , k )) . For 1 < p ≤
2, interpolation between (2.2) and (2.3) yields (cid:13)(cid:13) A m Λ (cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) Λ − α p with α p = + δ )(1 − / p ) −
1. When p ( d , k ) < p ≤
2, we have α p >
0, and hence, (cid:13)(cid:13) A m ∗ (cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) ≤ (cid:88) Λ= j (cid:38) (cid:13)(cid:13) A m Λ (cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) (cid:88) Λ= j (cid:38) Λ − α p (cid:46) . (2.4)The estimation of the major arc terms is more challenging, because an analogue of (2.3) does not holdfor A M Λ . Still, it is possible to establish a slightly weaker version of Proposition 2.1. The following result wasfirst established by Magyar [9], for d ≥ k , and then extended by the third author [6] in the present form. Proposition 2.2.
If k ≥ , d > k, dd − k < p ≤ , and Λ (cid:38) , one has (cid:13)(cid:13) A M Λ (cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) . (2.5)This proposition su ffi ces to establish the (cid:96) p -boundedness of the dyadic maximal functions A Λ (this is themain result of Magyar [9]), but falls just short of what is needed for an equally quick proof of Theorem 1.Instead, we use Theorem 2 to approximate A M ∗ by a bounded operator. Let M a / q λ denote the convolutionoperator on (cid:96) p ( Z d ) with Fourier multiplier (cid:91) M a / q λ ( ξ ) : = e q ( − λ a ) (cid:88) w ∈{± } d (cid:88) b ∈ Z d G ( q ; a , w b ) ψ ( q ξ − b ) (cid:103) d σ λ ( w ( ξ − q − b )) , and define M λ : = (cid:88) q ∈ N (cid:88) a ∈ Z ∗ q M a / q λ and M ∗ : = sup λ ∈ N | M λ | . In Section 4, we will handle the major arc approximations and prove the following proposition.6 roposition 2.3.
If k ≥ , d > k, and dd − k < p ≤ , then there exists an exponent β p = β p ( d , k ) > such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) A Mλ − M λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) Λ − β p . (2.6) Remark 2.1.
Theorem 2 is an immediate consequence of Propositions 2.1 and 2.3. Moreover, since when p = d ≥ k , inequality (2.6) holds for any β < / (2 k ) (see inequality (4.8) and the comments after it),the error bound (1.8) holds for all δ with 0 < δ < min { δ ( d , k ) , / (2 k ) } .When we sum (2.6) over dyadic Λ = j , we deduce that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ (cid:38) (cid:12)(cid:12) A Mλ − M λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) . Combining this bound and (2.4), we conclude that the (cid:96) p -boundedness of the maximal operator A ∗ followsfrom the (cid:96) p -boundedness of M ∗ . The following proposition, which we establish in Section 5, then completesthe proof of Theorem 1. Proposition 2.4.
If k ≥ , d > k, and dd − k < p ≤ , then M ∗ is bounded on (cid:96) p ( Z d ) . Our minor arc analysis splits naturally in two steps. The first step is a reduction to mean value estimatesrelated to Waring’s problem; for this we use a technique introduced in [1]. We then apply recent work onWaring’s problem to estimate the relevant mean values and to prove Proposition 2.1.
The reduction step is based on the following lemma, a special case of Lemma 7 in [1]. In the present form,the result is a slight variation of Lemma 4.2 in [6] and is implicit also in [11].
Lemma 3.1.
For λ ∈ N , let T λ be a convolution operator on (cid:96) ( Z d ) with Fourier multiplier given by (cid:98) T λ ( ξ ) : = (cid:90) B K ( θ ; ξ ) e ( − λθ ) d θ , where B ⊆ T is a measurable set and K ( · ; ξ ) ∈ L ( T ) is a kernel independent of λ . Further, for Λ ≥ , definethe dyadic maximal functions T ∗ f ( x ) = T ∗ ( x ; Λ ) : = sup λ ∈ [ Λ / , Λ ) | T λ f ( x ) | . Then (cid:107) T ∗ (cid:107) (cid:96) ( Z d ) → (cid:96) ( Z d ) ≤ (cid:90) B sup ξ ∈ T d | K ( θ ; ξ ) | d θ . (3.1)For a measurable set B ⊂ T , we have (cid:99) A Bλ ( ξ ) = λ − d / k (cid:90) B F N ( θ ; ξ ) e ( − λθ ) d θ , F N ( θ ; ξ ) : = d (cid:89) j = S N ( θ , ξ j ) with S N ( θ , ξ ) : = (cid:88) | n |≤ N e ( θ | n | k + ξ n ) . Thus, in the proof of Proposition 2.1, we apply (3.1) with K = F N and B = m . The supremum over ξ on theright side of (3.1) then stands in the way of a direct application of known results from analytic number theory.Our next lemma overcomes this obstacle; its proof is a variant of the argument leading to (12) in Wooley [18]. Lemma 3.2.
If k ≥ , l ≤ k − and s are natural numbers and B ⊆ T a measurable set, then (cid:90) B sup ξ ∈ T | S N ( θ , ξ ) | s d θ (cid:46) N l ( l + / (cid:90) B (cid:90) T l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ l n l + · · · + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d ξ d θ + . (3.2) Proof.
Define (cid:101) S N ( θ , ξ ) = (cid:88) ≤ n ≤ N e ( θ n k + ξ n ) . Since sup ξ ∈ T | S N ( θ , ξ ) | ≤ ξ ∈ T | (cid:101) S N ( θ , ξ ) | + , (3.3)it su ffi ces to establish (3.2) with (cid:101) S N in place of S N .Set H j = sN j . For h = ( h , . . . , h l ) ∈ Z l , we define a h ( θ ) = (cid:88) ≤ n ,..., n s ≤ N δ ( n ; h ) e ( θf s , k ( n )) , where δ ( n ; h ) = (cid:40) f s , j ( n ) = h j for all j = , . . . , l , . We have (cid:101) S N ( θ , ξ ) s = (cid:88) h ≤ H · · · (cid:88) h l ≤ H l a h ( θ ) e ( ξ h ) , so by applying the Cauchy–Schwarz inequality we deduce thatsup ξ | (cid:101) S N ( θ , ξ ) | s ≤ H · · · H l (cid:88) h ≤ H · · · (cid:88) h l ≤ H l | a h ( θ ) | . Hence, (cid:90) B sup ξ | (cid:101) S N ( θ , ξ ) | s d θ (cid:46) N l ( l + / (cid:90) B (cid:88) h ≤ H · · · (cid:88) h l ≤ H l a h ( θ ) a h ( θ ) d θ . (3.4)We have (cid:88) h ≤ H · · · (cid:88) h l ≤ H l a h ( θ ) a h ( θ ) = (cid:88) ≤ n , m ≤ N e (cid:0) θ ( f s , k ( n ) − f s , k ( m )) (cid:1) (cid:88) h ≤ H · · · (cid:88) h l ≤ H l δ ( n ; h ) δ ( m ; h ) .
8y orthogonality, (cid:88) h ≤ H · · · (cid:88) h l ≤ H l δ ( n ; h ) δ ( m ; h ) = (cid:88) h ≤ H · · · (cid:88) h l ≤ H l δ ( m ; h ) (cid:90) T l e (cid:18) l (cid:88) j = ξ j ( f s , j ( n ) − h j ) (cid:19) d ξ = (cid:90) T l e (cid:18) l (cid:88) j = ξ j ( f s , j ( n ) − f s , j ( m )) (cid:19) d ξ , since for a fixed m , the sum over h has exactly one term (in which h j = f s , j ( m )). Hence, (cid:88) h ≤ H · · · (cid:88) h l ≤ H l a h ( θ ) a h ( θ ) = (cid:90) T l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ l n l + · · · + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d ξ . (3.5)The lemma follows from (3.3)–(3.5). (cid:3) Remark 3.1.
With slight modifications, the argument of Lemma 3.2 yields also the following estimate (cid:90) B (cid:90) T | S N ( θ , ξ ) | s d ξ d θ (cid:46) N l ( l + / − (cid:90) B (cid:90) T l (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ l n l + · · · + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d ξ d θ + . We now recall several mean value estimates from the literature on Waring’s problem. The first is implicit inthe proof of Theorem 10 of Bourgain [3], which is a variant of a well-known lemma of Hua (see Lemma 2.5in [17]). The present result follows from eqn. (6.6) in [3]. We note that when l = k , the left side of (3.6) turnsinto Vinogradov’s integral J s , k ( N ) and the lemma turns into the main result of Bourgain, Demeter and Guth[4]. Lemma 3.3.
If k ≥ , ≤ l ≤ k and s ≥ l ( l + are natural numbers, then (cid:90) T (cid:90) T l − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ l − n l − + · · · + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d ξ d θ (cid:46) N s − l ( l + / + ε . (3.6)For small k , we will use another variant of Hua’s lemma due to Brüdern and Robert [5]. The following isa weak form of Lemma 5 in [5]. Lemma 3.4.
If k ≥ and ≤ l ≤ k are natural numbers, then (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l + d ξ d θ (cid:46) N l − l + + ε . (3.7)Note that by Remark 3.1 (with l − l ) and Lemma 3.3 we obtain a version of (3.7) with l ( l + l +
2. Together with Lemma 3.4, this observation yields the following bound.
Corollary 3.1.
If k ≥ and ≤ l ≤ k are natural numbers and r ≥ min { l ( l + , l + } , then (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r d ξ d θ (cid:46) N r − l − + ε . (3.8)9e also use a variant of Lemma 3.3 that provides extra savings when the integration over θ is restrictedto a set of minor arcs. Lemma 3.5, a slight modification of Theorem 1.3 in Wooley [18], improves on (3.6) inthe case l = k −
1. Here, m is the set of minor arcs defined at the beginning of §2. Lemma 3.5.
If k ≥ and s ≥ k ( k + are natural numbers, then (cid:90) m (cid:90) T k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ k − n k − + · · · + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d ξ d θ (cid:46) N s − k ( k − / − + ε . (3.9) Proof.
The main point in the proof of Theorem 2.1 in Wooley [18] is the inequality (see p. 1495 in [18]) (cid:90) m (cid:90) T k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ k − n k − + · · · + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s d ξ d θ (cid:46) N k − (log N ) s + J s , k (2 N ) . The lemma follows from this inequality and the Bourgain–Demeter–Guth bound for Vinogradov’s integral J s , k (2 N ) (the case l = k of (3.6)). (cid:3) Now we interpolate between the above bounds.
Lemma 3.6.
If k ≥ and ≤ l ≤ k − are natural numbers and r is real, withr ( k , l ) : = k − kl − min( l + l , l + k − l + ≤ r ≤ k + k , (3.10) then I r , k ( N ) : = (cid:90) m sup ξ ∈ T | S N ( θ , ξ ) | r d θ (cid:46) N r − k − δ ( r ) + ε , (3.11) where δ ( r ) is the linear function of r with values δ ( r ) = and δ ( k + k ) = .Proof. Let r = min { l + l , l + } . The hypothesis on r implies that r < r ≤ k + k , so we can find t ∈ [0 , r = tr + (1 − t )( k + k ). By Lemma 3.2 with l = I r , k ( N ) (cid:46) N (cid:90) T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:88) n = e ( θ n k + ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r d ξ d θ (cid:46) N r − l + ε . (3.12)On the other hand, Lemma 3.2 with l = k − I k ( k + , k ( N ) (cid:46) N ( k − k − / N k ( k + − k ( k − / − + ε = N k − + ε . (3.13)Using Hölder’s inequality and the above bounds, we get I r , k ( N ) (cid:46) I k ( k + , k ( N ) − t I r , k ( N ) t (cid:46) N (1 − t )( k − + ε ) N t ( r − l + ε ) (cid:46) N (1 − t )( k + k ) + tr N − (1 − t )( k + − tl + ε = N r − k − + t ( k − l + + ε . This inequality takes the form (3.11) with δ ( r ) = − t ( k − l + t depends linearly on r and t = r = k + k , δ ( r ) is a linear function of r with δ ( k + k ) =
1. The value of r ( k , l ) in (3.10) is the unique solutionof the linear equation δ ( r ) = (cid:3) .3 Proof of Proposition 2.1 By Lemma 3.1 and the arithmetic-geometric mean inequality, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) | A mλ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) ≤ N k − d (cid:90) m sup ξ ∈ T d | F N ( θ ; ξ ) | d θ ≤ N k − d I d , k ( N ) , (3.14)where I d , k ( N ) is the integral defined in (3.11). Thus, the proposition will follow, if we prove the inequality I d , k ( N ) (cid:46) N d − k − δ + ε (3.15)with δ = k δ ( d , k ).Let l ( k ) denote the value of l for which the maximum in the definition of d ( k ) is attained (recall (1.6)).When d ( k ) < d ≤ k + k , we may apply (3.11) with r = d and l = l to deduce (3.15) with δ = δ ( d ). We nowobserve that when d ≤ k + k , we have δ ( d ) = k δ ( d , k ) and that the hypothesis d > d ( k ) ensures that δ ( d ) > d > k + k , we enhance our estimates with the help of the L ∞ -bound for S N ( θ , ξ ) on the minor arcs:by combining a classical result of Weyl (see Lemma 2.4 in Vaughan [17]) and Theorem 5 in Bourgain [3], wehave sup ( θ , ξ ) ∈ m × T | S N ( θ , ξ ) | (cid:46) N − τ k + ε ,τ k being the quantity that appears in the definition of δ ( d , k ). Thus, when d > k + k , we have I d , k ( N ) (cid:46) N ( d − k − k )(1 − τ k ) + ε I k ( k + , k ( N ) (cid:46) N d − k − − ( d − k − k ) τ k + ε . We conclude that (3.15) holds with δ = + ( d − k − k ) τ k . (cid:3) We will proceed through a series of successive approximations to A a / q λ , which we will define by their Fouriermultipliers. Our approximations are based on the following major arc approximation for exponential sumsthat appears as part of Theorem 3 of Brüdern and Robert [5]. In this result and throughout the section, wewrite G ( q ; a , b ) : = q − (cid:88) x ∈ Z q e q ( ax k + bx ) and v N ( θ , ξ ) : = (cid:90) N e ( θ t k + ξ t ) dt . Lemma 4.1.
Let θ , ξ ∈ T , q ∈ N , a ∈ Z ∗ q , and b ∈ Z , and suppose that (cid:12)(cid:12)(cid:12)(cid:12) θ − aq (cid:12)(cid:12)(cid:12)(cid:12) ≤ kqN k − , (cid:12)(cid:12)(cid:12)(cid:12) ξ − bq (cid:12)(cid:12)(cid:12)(cid:12) ≤ q . Then (cid:88) n ≤ N e ( θ n k + ξ n ) = G ( q ; a , b ) v N ( θ − a / q , ξ − b / q ) + O (cid:0) q − / k + ε (cid:1) . Recall the definition of S N ( θ , ξ ) from Section 3.1. When a / q + θ lies on a major arc M ( a / q ) and ξ = b / q + η , with | η | ≤ / (2 q ), the above lemma yields S N ( a / q + θ , ξ ) = G ( q ; a , b ) v N ( θ , η ) + G ( q ; a , − b ) v N ( θ , − η ) + O (cid:0) q − / k + ε (cid:1) . (4.1)11e will use this approximation in conjunction with the following well-known bounds (see Theorems 7.1 and7.3 in Vaughan [17]): G ( q ; a , b ) (cid:46) q − / k + ε , v N ( θ , η ) (cid:46) N (1 + N | η | + N k | θ | ) − / k . (4.2)We will make use also of the inequality v N ( θ , η ) (cid:46) N (1 + N | η | ) − / , (4.3)which follows from the representation v N ( θ , η ) = k (cid:90) N k u / k − e ( θ u + η u / k ) du and the second-derivative bound for oscillatory integrals (see p. 334 in [14]). Proof of Proposition 2.3
The bulk of the work concerns the case p = (cid:100) A a / q λ ( ξ ) = λ − d / k e q ( − λ a ) (cid:90) M (0 / q ) F N ( a / q + θ ; ξ ) e ( − λθ ) d θ , the asymptotic (4.1) suggests that the following multiplier should provide a good approximation to (cid:100) A a / q λ ( ξ ): (cid:100) B a / q λ ( ξ ) : = λ − d / k e q ( − λ a ) (cid:90) M (0 / q ) G N ( θ ; ξ ) e ( − λθ ) d θ , where G N ( θ ; ξ ) : = d (cid:89) j = (cid:8) G ( q ; a , b j ) v N ( θ , η j ) + G ( q ; a , − b j ) v N ( θ , − η j ) (cid:9) , with b j the unique integer such that − ≤ b j − q ξ j < and η j = ξ j − b j / q . Let B a / q λ denote the operator on (cid:96) ( Z d ) with the above Fourier multiplier. To estimate the (cid:96) -error of approximation of A a / q λ by B a / q λ , we usethat when θ ∈ M (0 / q ), (4.1) and (4.2) yield | F N ( a / q + θ ; ξ ) − G N ( θ ; ξ ) | (cid:46) q − d / k + ε N d − (1 + N k | θ | ) (1 − d ) / k . Thus, if d > k +
1, we have (cid:90) M (0 / q ) sup ξ ∈ T d | F N ( a / q + θ ; ξ ) − G N ( θ ; ξ ) | d θ (cid:46) (cid:90) R q − d / k + ε N d − d θ (1 + N k | θ | ) ( d − / k (cid:46) q − d / k + ε N d − k − . Lemma 3.1 then yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) A a / q λ − B a / q λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) q − d / k + ε N − . (4.4)12ext, we approximate B a / q λ by the operator C a / q λ with multiplier (cid:100) C a / q λ ( ξ ) : = λ − d / k e q ( − λ a ) (cid:88) b ∈ Z d ψ ( q ξ − b ) (cid:90) M (0 / q ) G N ( θ ; ξ ) e ( − λθ ) d θ . By the localization of ψ , the above sum has at most one term in which b matches the integer vector thatappears in the definition of G N ( θ ; ξ ). Hence, G N ( θ ; ξ ) (cid:18) − (cid:88) b ∈ Z d ψ ( q ξ − b ) (cid:19) is supported on a set where ≤ | q ξ j − b j | ≤ for some j . For such j , by (4.3), v N ( θ , ξ j − b j / q ) (cid:46) ( qN ) / , and we conclude that | G N ( θ ; ξ ) | (cid:18) − (cid:88) b ∈ Z d ψ ( q ξ − b ) (cid:19) (cid:46) q / − d / k + ε N d − / (1 + N k | θ | ) (1 − d ) / k . So, when d > k +
1, Lemma 3.1 gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) B a / q λ − C a / q λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) (cid:90) R q / − d / k + ε N k − / d θ (1 + N k | θ | ) ( d − / k (cid:46) q / − d / k + ε N − / . (4.5)In our final approximation, we replace C a / q λ by the operator D a / q λ with multiplier (cid:100) D a / q λ ( ξ ) : = λ − d / k e q ( − λ a ) (cid:88) w ∈{± } d (cid:88) b ∈ Z d ψ ( q ξ − b ) G ( q ; a , w b ) J λ ( w ( ξ − q − b )) , where w b = ( w b , . . . , w d b d ) and J λ ( η ) : = (cid:90) R (cid:26) d (cid:89) j = v N ( θ ; η j ) (cid:27) e ( − λθ ) d θ . We remark that (cid:100) C a / q λ ( ξ ) can be expressed in a matching form, with J λ ( ξ − q − w b ) replaced by the analogousintegral over M (0 / q ). Thus, when d > k , we deduce from Lemma 3.1 and (4.2) that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) C a / q λ − D a / q λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) (cid:90) M (0 / q ) c q − d / k + ε N k d θ (1 + N k | θ | ) d / k (cid:46) q − + ε N − d / k . (4.6)Here, M (0 / q ) c denotes the complement of the interval M (0 / q ) in R .13inally, we note that D a / q λ is really M a / q λ . Indeed, by the discussion on p. 498 in Stein [14] (see alsoLemma 5 in Magyar [10]), we have J λ ( η ) = (cid:90) R (cid:90) R d [0 , N ] d ( t ) e ( η · t ) e ( θ ( f ( t ) − λ )) d t d θ = λ d / k − (cid:90) R d [0 , N ] d ( t ) e ( η · t ) d σ λ ( t ) = λ d / k − (cid:103) d σ λ ( η ) , (4.7)since the surface measure d σ λ is supported in the cube [0 , N ] d . Combining this observation and (4.4)–(4.6),and summing over a , q , we conclude that when d > k , (cid:88) q ≤ N (cid:88) a ∈ Z ∗ q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) A a / q λ − M a / q λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) N − γ + ε , (4.8)with γ = min (cid:0) d / k − , (cid:1) >
0. In particular, when d ≥ k , we have γ = .We can now finish the proof of Proposition 2.3. For brevity, we write p = dd − k . From (4.8), we obtainthat (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) A Mλ − M λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) ( Z d ) → (cid:96) ( Z d ) (cid:46) N − γ + ε . (4.9)On the other hand, we know from Propositions 2.2 and 2.4 that both A M Λ and M ∗ are bounded on (cid:96) p ( Z d ) when p < p ≤
2; hence, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ [ Λ / , Λ ) (cid:12)(cid:12) A Mλ − M λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) . (4.10)When p ∈ ( p ( d , k ) , p = p + ε for a su ffi ciently small ε > β p > γ ( p − p ) / ( k p (2 − p )). (cid:3) In this section, we prove Proposition 2.4. First, we obtain L p ( R d )-bounds for the maximal function of thecontinuous surface measures d σ λ . Lemma 5.1.
If k ≥ , d > k and p > d − k d − k , then for all f ∈ L p ( R d ) , (cid:13)(cid:13)(cid:13)(cid:13) sup λ > | f ∗ d σ λ | (cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) (cid:46) (cid:107) f (cid:107) L p ( R d ) . (5.1) Proof.
We will deduce the lemma from a result of Rubio de Francia – Theorem A in [13] – which reduces(5.1) to bounds for the Fourier transform of the measure d σ λ . First, we majorize the measure d σ λ by asmooth one. By the choice of normalization of d σ λ , we have f ∗ d σ λ ( x ) = (cid:90) R d f ( x − y ) d σ λ ( y ) = (cid:90) R d f ( x − t y ) d σ ( y ) , where t = λ / k . Let φ be a smooth function supported in (cid:2) − , (cid:3) and such that [0 , ( x ) ≤ φ ( x ), and write φ ( x ) = φ ( x ) · · · φ ( x d ). Since d σ is supported inside the unit cube [0 , d , we have (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R d f ( x − t y ) d σ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) R d | f ( x − t y ) | φ ( y ) d σ ( y ) = : (cid:90) R d | f ( x − t y ) | d µ ( y ) , d σ is the surface measure on the smooth manifold x k + · · · + x kd =
1. By Rubio de Francia’s theorem,the maximal function A t f ( x ) : = sup t > (cid:90) R d | f ( x − t y ) | d µ ( y )is bounded on L p ( R d ), provided that (cid:101) µ ( ξ ) (cid:46) ( | ξ | + − a for some a > min (cid:26) p − , (cid:27) . (5.2)Thus, the lemma will follow, if we establish (5.2) with a = d / k − d > k + a > / d > k .We now turn to (5.2). Similarly to (4.7), we have (cid:101) µ ( ξ ) = (cid:90) R (cid:90) R d φ ( x ) e ( ξ · x ) e ( θ ( f ( x ) − d x d θ = (cid:90) R (cid:26) d (cid:89) j = v φ ( θ , ξ j ) (cid:27) e ( − θ ) d θ , where v φ ( θ , ξ ) : = (cid:90) R φ ( x ) e ( θ x k + ξ x ) dx . By the corollary on p. 334 of Stein [14], we have v φ ( θ , ξ ) (cid:46) (1 + | θ | ) − / k , (5.3)uniformly in ξ . On the other hand, if k k | θ | ≤ | ξ | , we have (cid:12)(cid:12)(cid:12)(cid:12) ddx (cid:0) ξ − θ x k + x (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ≥ , (cid:12)(cid:12)(cid:12)(cid:12) d j dx j (cid:0) ξ − θ x k + x (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) (cid:46) j φ . Hence, Proposition VIII.1 on p. 331 of Stein [14] yields v φ ( θ , ξ ) (cid:46) M (1 + | ξ | ) − M (5.4)for any fixed M ≥ j , 1 ≤ i ≤ d , such that | ξ | ≤ d | ξ i | and set θ = | ξ i | / ( k k ). We apply (5.3) to thetrigonometric integrals v φ ( θ , ξ j ), j (cid:44) i , and to v φ ( θ , ξ i ) when | θ | > θ ; we apply (5.4) to v φ ( θ , ξ i ) when | θ | ≤ θ .From these bounds and the integral representation for (cid:101) µ ( ξ ), we obtain (cid:101) µ ( ξ ) (cid:46) (cid:90) | θ |≤ θ (1 + | ξ i | ) − M (1 + | θ | ) ( d − / k d θ + (cid:90) | θ | > θ d θ (1 + | θ | ) d / k (cid:46) (1 + | ξ i | ) − M + (1 + | ξ i | ) − d / k (cid:46) (1 + | ξ | ) − d / k , provided that M ≥ d / k − d > k +
1. This completes the proof. (cid:3)
Proof of Proposition 2.4
To prove Proposition 2.4, it su ffi ces to prove that (uniformly in a and q ) (cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ N (cid:12)(cid:12) M a / q λ (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) q − dk (2 − / p ) + ε , dd − k < p ≤ d > k . The proposition then follows by summing over a and q (the hypothesis on p ensures that the resulting series over q is convergent).Fix q ∈ N and a ∈ Z ∗ q and write ψ ( ξ ) = ψ ( ξ /
2) (so that ψ = ψψ ). We borrow a trick from Magyar,Stein and Wainger [11] to express (cid:91) M a / q λ ( ξ ) as a linear combination of Fourier multipliers that separate thedependence on λ and from the dependence on a / q : (cid:91) M a / q λ ( ξ ) = e q ( − λ a ) (cid:88) w ∈{± } d (cid:88) b ∈ Z d ψ ( q ξ − b ) G ( q ; a , w b ) (cid:103) d σ λ ( w ( ξ − q − b )) = e q ( − λ a ) (cid:88) w ∈{± } d (cid:18) (cid:88) b ∈ Z d ψ ( q ξ − b ) G ( q ; a , w b ) (cid:19)(cid:18) (cid:88) b ∈ Z d ψ ( q ξ − b ) (cid:103) d σ λ ( w ( ξ − q − b )) (cid:19) = : e q ( − λ a ) (cid:88) w ∈{± } d (cid:100) S a / q w ( ξ ) (cid:100) T q λ , w ( ξ ) . Since w takes on precisely 2 d values for each a / q , it su ffi ces to prove that (cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ N (cid:12)(cid:12) T q λ , w ◦ S a / q w (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) q − dk (2 − / p ) + ε , (5.5)uniformly for w ∈ {− , } d . To prove (5.5), we will first bound the maximal function over the ‘Archimedean’multipliers T q λ , w , and then we will bound the non-Archimedean multipliers S a / q w . This is possible because S a / q w is independent of λ ∈ N .For d > k and p > d − k d − k , Lemma 5.1 and Corollary 2.1 of [11] (the ‘Magyar–Stein–Wainger transferenceprinciple’) give the bound (cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ N (cid:12)(cid:12) T q λ , w g (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) (cid:46) (cid:107) g (cid:107) (cid:96) p ( Z d ) , (5.6)with an implicit constant independent of q and w . We now observe that S a / q w does not depend on λ and apply(5.6) with g = S a / q w f to find that (cid:13)(cid:13)(cid:13)(cid:13) sup λ ∈ N (cid:12)(cid:12) T q λ , w ◦ S a / q w (cid:12)(cid:12)(cid:13)(cid:13)(cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) (cid:13)(cid:13) S a / q w (cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) , (5.7)under the same assumptions on d and p which we note are weaker than the hypotheses of the proposition.Finally, observe that G ( q ; a , b ) is a q -periodic function with Z q -Fourier transform equal to (cid:88) b ∈ Z q e q ( − mb ) G ( q ; a , b ) = q − (cid:88) x ∈ Z q e q ( ax k ) (cid:88) b ∈ Z q e q ( b ( x − m )) = e q ( am k ) , for each m ∈ Z q . Hence, we may apply Proposition 2.2 in [11] and the bound (4.2) for G ( q ; a , b ) to deducethat (cid:13)(cid:13) S a / q w (cid:13)(cid:13) (cid:96) p ( Z d ) → (cid:96) p ( Z d ) (cid:46) q − dk (2 − / p ) + ε . (5.8)The desired inequality (5.5) follows immediately from (5.7) and (5.8). (cid:3) Acknowledgments
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