Incomplete gauss sums modulo primes

Incomplete gauss sums modulo primes Abstract We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov’s method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov’s mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range N≥q2k−1/2+O(k−3/2). 1. Introduction The estimation of exponential sums of the form ∑M<n≤M+Ne2πif(n), where f is a polynomial of large degree and is a common problem in number theory with a wide range of arithmetic consequences. This problem has been considered by a number of previous authors, see [4, 12, 18, 21] for recent progress on the estimation of such sums. See also [9] and [13, Chapter 8] for a brief overview of current results and techniques. Let a and q be integers with (a,q)=1. In this paper, we consider the problem of estimating sums of the form ∑M<n≤M+Neq(ank), (1.1) where eq(z)=e2πiz/q. Sums of the form ∑1≤n≤Ne(αnk), where e(z)=e2πiz, arise from an application of the circle method to Waring’s problem, which asks for the smallest integer m such that all sufficiently large integers are the sum of at most mkth powers, see for example [20, 24]. When considering the minor arcs in Waringʼs problem, one uses a bound of the form ∣∑M<n≤M+Ne(αnk)∣≤N1−1/ρ, (1.2) where ρ depends on the integer k, a rational approximation a/q to α and the size of N relative to q. The order of ρ as a function of k is an important factor in applications and it is desirable for ρ to grow as slowly as possible for large k. For large k, current methods for producing the sharpest bound for (1.2) reduce the problem to estimating the number of solutions to a certain system of equations known as Vinogradov’s mean value theorem, which we describe below. For these values of k, the best known estimate for (1.2) relies on recent results of Bourgain et al. [5] (see for example [16, Lemma 2.1] and [9, Eq. (4.5)]) and may be stated as follows. Let k≥3 and suppose that ∣α−aq∣≤1q2. If N≤q≤Nk−1, then we have ∣∑M<n≤M+Ne(αnk)∣≤N1−1/k(k−1)+o(1). (1.3) For certain α, the bound (1.3) is sharpest known only when N is small relative to q. For example, when q is prime, one may use the Weil bounds to show ∣∑M<n≤M+Neq(ank)∣≪q1/2logq, (1.4) which is better than (1.3) in the range N≥q1/2+1/k2+O(1/k3). For integers k,r and V, we let Jr,k(V) count the number of solutions to the system of equations v1j+⋯+vrj=vr+1j+⋯+v2rj,j=1,…,k, (1.5) with variables satisfying 1≤v1,…,v2r≤V. Bounds for Jr,k(V) are usually referred to as Vinogradov’s mean value theorem and typically take the shape Jr,k(V)≤(1+Vr−k(k+1)/2)Vr+ε, (1.6) for any fixed ε>0. The main conjecture for Jr,k(V) is the statement that (1.6) holds for all integers r,k and V. Significant progress concerning bounds for Jr,k(V) has been made by Wooley [22, 23, 25, 26], and in particular Wooley [27] has proven the main conjecture for Jr,k(V) in the case k=3. More recently, Bourgain et al. [5] have proven the main conjecture for Jr,k(V) when k>3. Combining the results of Wooley [27] for the case k=3 with those of Bourgain et al. for the case k>3, for any integers r,k and V, we have Jr,k(V)≤(1+Vr−k(k+1)/2)Vr+ε, (1.7) for any fixed ε>0. We also refer the reader to a recent paper of Wooley [28] for new results concerning Vinogradov’s mean value theorem and related systems of equations. In this paper we obtain a new bound for the sums (1.1) when q is prime. Our argument falls under the framework of Vinogradov’s method which we use to reduce the problem to bounding the number of solutions of two systems of congruences. The first concerns the number of solutions to the system v1j+⋯+vrj≡vr+1j+⋯+v2rjmodq,j=1,…,k, (1.8) which has been considered by Karatsuba [14], who attributes the problem to Korobov. We bound the number of solutions to this system by reducing to Vinogradov’s mean value theorem and applying results of Wooley [27] and Bourgain et al. [5]. The second system of congruences (see Lemma 3.4) does not appear to have been considered before and we use some ideas based on Mordell [17]. We also note that a related system of equations has been considered by Arkhipov and Karatsuba [1]. The sums (1.1) also arise in considering analogues of Waring’s problem in prime fields with variables restricted to boxes, see for example [6, 7]. See also [3, 8, 19] for related problems. 2. Main results Our main result is as follows. Theorem 2.1. Let k≥3be an integer, qa prime number, aan integer with (a,q)=1and suppose Mand Nare integers with Nsatisfying N≤q1/2+1/(k1/2+1). (2.1)Then we have ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqo(1). We first note that the bound of Theorem 2.1 is non-trivial in the range q(1+ε)/(k−2)1/2≤N, (2.2) and, in this case, we have a bound of the form ∣∑M<n≤M+Neq(ank)∣≤N1−ρ+o(1), where ρ=2ε(1+ε)k(k+1). Comparing Theorem 2.1 with the bound (1.3), we see that Theorem 2.1 provides an improvement over (1.3) in the range N≥q2(k−1)/(k−3)(k−2)1/2=q2k−1/2+O(k−3/2). 3. Preliminary results We first prove the following general inequality for systems of congruences. Lemma 3.1. Suppose qis prime and t,mand ℓare integers. Let X⊆(Z/qZ)t,be a set and f={fi}i=1ma sequence of functions on X, fi:X→Z/qZ,i=1,…,m.Let σ={σi}i=12ℓ, be a sequence of numbers with each σi≢0modqand λ={λj}j=1ma sequence of numbers with each λj∈Z/qZ. We let Im,ℓ(f,X,σ,λ)denote the number of solutions to the system of congruences σ1fj(x1)+⋯+σ2ℓfj(x2ℓ)≡λjmodq,j=1,…,m, (3.1)with variables x1,…,x2ℓsatisfying x1,…,x2ℓ∈X.In the special case that σ={(−1)i}i=12ℓand each λj=0, we write Im,ℓ(f,X,{(−1)i}i=12ℓ,0)=Im,ℓ(f,X).For any X,f,σ and λas above, we have Im,ℓ(f,X,σ,λ)≤Im,ℓ(f,X). Proof Expanding the system (3.1) into additive characters, we see that Im,ℓ(f,X,σ,λ)=1qm∑1≤ah≤q1≤h≤m∏1≤i≤2ℓ(∑xi∈Xeq(σi(∑j=1majfj(xi))))eq(−∑j=1majλj), so that writing f(a1,…,am)=∑x∈Xeq(∑j=1majfj(x)), the above implies Im,ℓ(f,X,σ,λ)=1qm∑1≤ah≤q1≤h≤m∏1≤i≤2ℓf(σia1,…,σiam)eq(−∑j=1majλj), which we may bound by Im,ℓ(f,X,σ,λ)≤1qm∑1≤ah≤q1≤h≤m∏1≤i≤2ℓ∣f(σia1,…,σiam)∣. An application of Hölder’s inequality gives Im,ℓ(f,X,σ,λ)≤∏i=12ℓ(1qm∑1≤ah≤q1≤h≤m∣f(σia1,…,σiam)∣2ℓ)1/2ℓ, and hence Im,ℓ(f,X,σ,λ)≤∏i=12ℓ(1qm∑1≤ah≤q1≤h≤m∣f(a1,…,am)∣2ℓ)1/2ℓ=1qm∑1≤ah≤q1≤h≤m∣f(a1,…,am)∣2ℓ. The result follows since the term 1qm∑1≤ah≤q1≤h≤m∣f(a1,…,am)∣2ℓ, counts the number of solutions to the system of congruences ∑i=12ℓ(−1)ifj(xi)≡0,j=1,…,m, with variables x1,…,x2ℓ satisfying x1,…,x2ℓ∈X. □ The proof of the following uses the bound (1.7). Lemma 3.2. For integers k,r,Vand q, we let Jr,k(V;q)count the number of solutions to the system of congruences v1j+⋯+vrj≡vr+1j+⋯+v2rjmodq,j=1,…,k, (3.2)with variables satisfying 1≤v1,…,v2r≤V.Let 1≤m<kbe an integer and suppose Vsatisfies q1/m≪V<q1/mr. (3.3)If r≥k(k+1)/2, we have Jr,k(V;q)≤V2r−km+m(m−1)/2+o(1). Proof For integers λm+1,…,λr, we let Jr,k(V,λm+1,…,λk) denote the number of solutions to the system of equations v1j+⋯+vrj=vr+1j+⋯+v2rj,j=1,…,m, v1j+⋯+vrj=vr+1j+⋯+v2rj+λj,j=m+1,…,k, with variables v1,…,v2r satisfying 1≤v1,…,v2r≤V. Expressing Jr,k(V,λm+1,…,λk) via additive characters, we see that Jr,k(V,λm+1,…,λk)=∫01⋯∫01∣∑1≤v≤Ve(α1v+⋯+αkvk)∣2r×e(αm+1λm+1+⋯+αkλk)dα1…dαk, where e(z)=e2πiz, and hence Jr,k(V,λm+1,…,λk)≤∫01⋯∫01∣∑1≤v≤Ve(α1v+⋯+αkvk)∣2rdα1…dαk. This implies that Jr,k(V,λm+1,…,λk)≤Jr,k(V). (3.4) Using the assumption (3.3), we have Jr,k(V;q)=∑∣λj∣≤rVj/qm+1≤j≤kJr,k(V,λm+1q,…,λkq), and hence by (3.4) Jr,k(V;q)≪(∏j=m+1kVjq)Jr,k(V). Since r≥k(k+1)/2, an application of (1.7) gives Jr,k(V;q)≪(∏j=m+1kVjq)V2r−k(k+1)/2+o(1), which on recalling (3.3) simplifies to Jr,k(V;q)≪V2r−km+m(m−1)/2+o(1). □ For a proof of the following, see [10]. See also [2, 11, 15] for related and more precise results. Lemma 3.3. Let qbe prime and suppose M,Nand Uare integers with Nand Usatisfying NU≤q,U≤N. The number of solutions to the congruence n1u1≡n2u2modq, with variables satisfying M<n1,n2≤M+N,1≤u1,u2≤U, is bounded by O(NUlogq). Lemma 3.4. Let qbe prime and M,Nand Uintegers with Nand Usatisfying NU≤q,U≤N.Let ℓ≥1be an integer and suppose kis an integer satisfying 2ℓ≤k.Let Ik,ℓ(N,U)denote the number of solutions to the system of congruences n1ju1k−j+⋯+nℓjuℓk−j≡nℓ+1juℓ+1k−j+⋯+n2ℓju2ℓk−jmodq,j=0,…,2ℓ−1, (3.5)in variables n1,…,n2ℓ,u1,…,u2ℓsatisfying M<n1,…,n2ℓ≤M+N,1≤u1,…,u2ℓ≤U. (3.6)Then we have Ik,ℓ(N,U)≪(NU)ℓ(logq)ℓ−1. Proof We fix an integer k and proceed by induction on ℓ. Considering first the case ℓ=1, we recall that Ik,1(N,U) counts the number of solutions to the congruences u1k≡u2kmodqandu1k−1n1≡u2k−1n2modq, in variables M≤n1,n2≤M+N,1≤u1,u2≤U. Fixing a value of u1 determines u2 with at most k choices and fixing a value of n1 determines n2 with at most 1 choice. This implies that Ik,1(N,U)≪NU. We next assume that for integer ℓ satisfying ℓ−1≥1 and 2ℓ≤k, we have Ik,ℓ−1(N,U)≪(NU)ℓ−1(logq)ℓ−2, (3.7) and we aim to show that Ik,ℓ(N,U)≪(NU)ℓ(logq)ℓ−1. (3.8) Suppose the points n=(n1,…,n2ℓ) and u=(u1,…,u2ℓ) are a solution to (3.5) satisfying (3.6). Let H(u)⊆Fq2ℓ, denote the hyperplane H(u)={(z1,…,z2ℓ)∈Fq2ℓ:z1u1k+⋯−z2ℓu2ℓk≡0modq}. Considering the system (3.5), we see that the 2ℓ points pj(n,u)=((n1u1)j,…,(n2ℓu2ℓ)j),j=0,…,2ℓ−1, all lie on H(u). Since H(u) has dimension 2ℓ−1, this implies that there exists some non-trivial linear relation among the pj(n,u). In particular, there exists α0,…,α2ℓ−1∈Fq, with at least one αi≢0modq such that α0p0(n,u)+⋯+α2ℓ−1p2ℓ−1(n,u)≡0modq. (3.9) Equation (3.9) implies that the Vandermonde matrix of the points n1u1,…,n2ℓu2ℓ, is singular mod q, and hence there exist integers 1≤r,s≤2ℓ with r≠s such that nrur≡nsusmodq. (3.10) Letting Ik,ℓ(N,U,r,s) denote the number of solutions to the system (3.5) with variables satisfying (3.6) subject to the further condition (3.10), since the pair r,s can take at most 4ℓ2 values, we see that Ik,ℓ(N,U)≪Ik,ℓ(N,U,r,s), (3.11) for some pair r,s with r≠s. Considering Ik,ℓ(N,U,r,s), there exists some sequence of numbers σi∈{−1,1} for 1≤i≤2ℓ such that Ik,ℓ(N,U,r,s) is equal to the number of solutions to the system of congruences ∑1≤i≤2ℓi≠r,sσinijuik−j≡σrnrjurk−j+σsnsjusk−jmodq,j=0,…,2ℓ−1, (3.12) with variables satisfying (3.6) and (3.10). For each fixed nr,ns,ur,us, we let I0(nr,ns,ur,us) count the number of solutions to the system (3.12) with variables satisfying M<ni≤M+N,1≤ui≤U,i=1,…,2ℓ,i≠r,s, (3.13) so that the above implies Ik,ℓ(N,U)≪∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodqI0(nr,ns,ur,us). (3.14) Letting I0′ denote the number of solutions to the system of congruences n1ju1k−j+⋯+nℓ−1juℓ−1k−j≡nℓjuℓk−j+⋯+n2ℓ−2ju2ℓ−2k−jmodq,j=0,…,2ℓ−1, (3.15) in variables n1,…,n2ℓ−2,u1,…,u2ℓ−2 satisfying M<n1,…,n2ℓ−2≤M+N,1≤u1,…,u2ℓ−2≤U, (3.16) an application of Lemma 3.1 gives I0(nr,ns,ur,us)≤I0′, (3.17) for any nr,ns,ur and us. Considering only equations corresponding to j=0,…,2(ℓ−1)−1, in the system (3.15), we see that I0′≤Ik,ℓ−1(N,U), and hence by (3.17) I0(nr,ns,ur,us)≪Ik,ℓ−1(N,U). Combining the above with (3.14), we get Ik,ℓ(N,U)≪(∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodq1)Ik,ℓ−1(N,U). Since the term ∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodq1, counts the number of solutions to the congruence nrus≡nsurmodq, in variables nr,ns,ur,us satisfying M<nr,ns≤M+N,1≤ur,us≤U, an application of Lemma 3.3 gives ∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodq1≪NUlogq, and hence Ik,ℓ(N,U)≪Ik,ℓ−1(N,U)NUlogq. Combining the above with (3.7), we see that Ik,ℓ(N,U)≪(NU)ℓ(logq)ℓ−1, which establishes (3.8).□ 4. Proof of Theorem 2.1 We proceed by induction on N and fix an arbitrarily small ε. Since the bound of Theorem 2.1 is trivial provided N≤q1/(k−2)1/2, this forms the basis of the induction. We formulate our induction hypothesis as follows. Let M be an arbitrary integer and suppose for all integers K≤N−1 we have ∣∑M<n≤M+Keq(ank)∣≤(q1/(k−2)1/2K)2/k(k+1)Kqε, uniformly over M. Using the above hypothesis, we aim to show ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqε. (4.1) We define the integers ℓ,m and r by ℓ=⌊k2⌋, (4.2) m=⌈2ℓ⌉, (4.3) r=k(k+1)2, (4.4) and define the integers U and V by U=⌊rN2q1/m⌋,V=⌊q1/m2r⌋. (4.5) We first note that UV≤N4. (4.6) Let 1≤u≤U and 1≤v≤V be integers and write ∑M<n≤M+Neq(ank)=∑M−uv<n≤M+N−uveq(a(n+uv)k)=∑M<n≤M+Neq(a(n+uv)k)+∑M−uv<n≤Meq(a(n+uv)k)−∑M+N−uv<n≤M+Neq(a(n+uv)k). Averaging the above over 1≤u≤U and 1≤v≤V, using (4.6) and applying our induction hypothesis, we see that ∣∑M<n≤M+Neq(ank)∣≤∣W∣UV+12(q1/(k−2)1/2N)2/k(k+1)Nqε, (4.7) where W=∑1≤u≤U∑1≤v≤V∑M<n≤M+Neq(a(n+uv)k). We have ∣W∣≤∑1≤v≤V∣∑1≤u≤U∑M<n≤M+Neq(a∑j=0k(kj)njuk−jvk−j)∣, hence by Hölder’s inequality ∣W∣ℓ≤Vℓ−1∑1≤v≤V∣∑1≤u≤U∑M<n≤M+Neq(a∑j=0k(kj)njuk−jvk−j)∣ℓ, so that for some complex numbers θv with ∣θv∣=1, we have ∣W∣ℓ≤Vℓ−1∑1≤ui≤U1≤i≤ℓ∑M<ni≤M+N1≤i≤ℓ∣∑1≤v≤Vθveq(a∑j=0k−1(kj)(n1ju1k−j+⋯+nℓjuℓk−j)vk−j)∣. Let W0=∑1≤ui≤U∑M<ni≤M+N∣∑1≤v≤Vθveq(a∑j=0k−1(kj)(n1ju1k−j+⋯+nℓjuℓk−j)vk−j)∣, so that ∣W∣ℓ≪Vℓ−1W0. (4.8) Let I(λ0,…,λk−1) denote the number of solutions to the system of congruences (kj)(n1ju1k−j+⋯+nℓjuℓk−j)≡λjmodq,j=0,…,k−1, in variables n1,…,nℓ,u1,…,uℓ satisfying M<n1,…,nℓ≤M+N,1≤u1,…,uℓ≤U. The above implies that W0=∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)∣∑1≤v≤Vθveq(a∑j=0k−1λjvk−j)∣. Two applications of Hölder’s inequality gives W02r≤(∑λ0,…,λk−1=0q−1I(λ0,…,λk−1))2r−2(∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2)×(∑λ0,…,λk−1=0q−1∣∑1≤v≤Vθveq(∑j=0k−1λjvk−j∣)2r). (4.9) The term ∑λ0,…,λk−1=0q−1∣∑1≤v≤Vθveq(∑j=0k−1λjvk−j)∣2r, is bounded by qk times the number of solutions to the system of congruences v1j+⋯+vrj≡vr+1j+⋯+v2rjmodq,j=1,…,k, in variables v1,…,v2r satisfying 1≤v1,…,v2r≤V. Recalling the choice of V in (4.5) and applying Lemma 3.2, we see that ∑λ0,…,λk−1=0q−1∣∑1≤v≤Vθveq(∑j=0k−1λjvk−j)∣2r≤qkJr,k(V;q)≤qkV2r−mk+m(m−1)/2+o(1). (4.10) We have ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)=(NU)ℓ, (4.11) and the term ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2, is equal to the number of solutions to the system of congruences n1ju1k−j+⋯+nℓjuℓk−j≡nℓ+1juℓ+1k−j+⋯+n2ℓju2ℓk−jmodq,j=0,…,k−1, (4.12) in variables n1,…,n2ℓ,u1,…,u2ℓ satisfying M<n1,…,n2ℓ≤M+N,1≤u1,…,u2ℓ≤U. Recalling (4.2) and considering only equations corresponding to j=0,…,2ℓ−1 in (4.12), we see that ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2≤Ik,ℓ(N,U), where Ik,ℓ(N,U) is defined as in Lemma 3.4. Recalling (2.2), we have logq≪logN=No(1), and hence ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2≤Ik,ℓ(N,U)≤(NU)ℓ+o(1). (4.13) Combining (4.9) (4.10), (4.11) and (4.13) gives W02r≤qk+o(1)(NU)ℓ(2r−1)V2r−mk+m(m−1)/2, and hence by (4.8) ∣W∣ℓ≪qk/2r+o(1)Vℓ−mk/2r+m(m−1)/4r(NU)ℓ(1−1/2r). By (4.7) we have ∣∑M<n≤M+Neq(ank)∣≤(qkVmk−m(m−1)/2)1/2rℓN1−1/2rqo(1)U1/2r+12(q1/(k−2)1/2N)2/k(k+1)Nqε, which on recalling the choice of U and V in (4.5), the above simplifies to ∣∑M<n≤M+Neq(ank)∣≤q((m−1)/2ℓ+1/m)/2r+o(1)N1−1/r+12(q1/(k−2)1/2N)2/k(k+1)Nqε. Recalling the choice of ℓ, m and r in (4.2), (4.3) and (4.4), we get ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqo(1)+12(q1/(k−2)1/2N)2/k(k+1)Nqε, and hence ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqε, by taking the term o(1) in qo(1) to be sufficiently small. Acknowledgement The author would like to thank Igor Shparlinski for bringing the paper of Karatsuba [14] to the author’s attention, for explaining the Russian version of Karatsuba’s paper and for his comments on a preliminary version of the current paper. References 1 G. I. Arkhipov and A. A. 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Wooley , The cubic case of the main conjecture in Vinogradov’s mean value theorem , Adv. Math. 294 ( 2016 ), 532 – 561 . Google Scholar CrossRef Search ADS 28 T. D. Wooley , Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, arXiv:1708.01220. © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

Incomplete gauss sums modulo primes

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Abstract

Abstract We obtain a new bound for incomplete Gauss sums modulo primes. Our argument falls under the framework of Vinogradov’s method which we use to reduce the problem under consideration to bounding the number of solutions to two distinct systems of congruences. The first is related to Vinogradov’s mean value theorem, although the second does not appear to have been considered before. Our bound improves on current results in the range N≥q2k−1/2+O(k−3/2). 1. Introduction The estimation of exponential sums of the form ∑M<n≤M+Ne2πif(n), where f is a polynomial of large degree and is a common problem in number theory with a wide range of arithmetic consequences. This problem has been considered by a number of previous authors, see [4, 12, 18, 21] for recent progress on the estimation of such sums. See also [9] and [13, Chapter 8] for a brief overview of current results and techniques. Let a and q be integers with (a,q)=1. In this paper, we consider the problem of estimating sums of the form ∑M<n≤M+Neq(ank), (1.1) where eq(z)=e2πiz/q. Sums of the form ∑1≤n≤Ne(αnk), where e(z)=e2πiz, arise from an application of the circle method to Waring’s problem, which asks for the smallest integer m such that all sufficiently large integers are the sum of at most mkth powers, see for example [20, 24]. When considering the minor arcs in Waringʼs problem, one uses a bound of the form ∣∑M<n≤M+Ne(αnk)∣≤N1−1/ρ, (1.2) where ρ depends on the integer k, a rational approximation a/q to α and the size of N relative to q. The order of ρ as a function of k is an important factor in applications and it is desirable for ρ to grow as slowly as possible for large k. For large k, current methods for producing the sharpest bound for (1.2) reduce the problem to estimating the number of solutions to a certain system of equations known as Vinogradov’s mean value theorem, which we describe below. For these values of k, the best known estimate for (1.2) relies on recent results of Bourgain et al. [5] (see for example [16, Lemma 2.1] and [9, Eq. (4.5)]) and may be stated as follows. Let k≥3 and suppose that ∣α−aq∣≤1q2. If N≤q≤Nk−1, then we have ∣∑M<n≤M+Ne(αnk)∣≤N1−1/k(k−1)+o(1). (1.3) For certain α, the bound (1.3) is sharpest known only when N is small relative to q. For example, when q is prime, one may use the Weil bounds to show ∣∑M<n≤M+Neq(ank)∣≪q1/2logq, (1.4) which is better than (1.3) in the range N≥q1/2+1/k2+O(1/k3). For integers k,r and V, we let Jr,k(V) count the number of solutions to the system of equations v1j+⋯+vrj=vr+1j+⋯+v2rj,j=1,…,k, (1.5) with variables satisfying 1≤v1,…,v2r≤V. Bounds for Jr,k(V) are usually referred to as Vinogradov’s mean value theorem and typically take the shape Jr,k(V)≤(1+Vr−k(k+1)/2)Vr+ε, (1.6) for any fixed ε>0. The main conjecture for Jr,k(V) is the statement that (1.6) holds for all integers r,k and V. Significant progress concerning bounds for Jr,k(V) has been made by Wooley [22, 23, 25, 26], and in particular Wooley [27] has proven the main conjecture for Jr,k(V) in the case k=3. More recently, Bourgain et al. [5] have proven the main conjecture for Jr,k(V) when k>3. Combining the results of Wooley [27] for the case k=3 with those of Bourgain et al. for the case k>3, for any integers r,k and V, we have Jr,k(V)≤(1+Vr−k(k+1)/2)Vr+ε, (1.7) for any fixed ε>0. We also refer the reader to a recent paper of Wooley [28] for new results concerning Vinogradov’s mean value theorem and related systems of equations. In this paper we obtain a new bound for the sums (1.1) when q is prime. Our argument falls under the framework of Vinogradov’s method which we use to reduce the problem to bounding the number of solutions of two systems of congruences. The first concerns the number of solutions to the system v1j+⋯+vrj≡vr+1j+⋯+v2rjmodq,j=1,…,k, (1.8) which has been considered by Karatsuba [14], who attributes the problem to Korobov. We bound the number of solutions to this system by reducing to Vinogradov’s mean value theorem and applying results of Wooley [27] and Bourgain et al. [5]. The second system of congruences (see Lemma 3.4) does not appear to have been considered before and we use some ideas based on Mordell [17]. We also note that a related system of equations has been considered by Arkhipov and Karatsuba [1]. The sums (1.1) also arise in considering analogues of Waring’s problem in prime fields with variables restricted to boxes, see for example [6, 7]. See also [3, 8, 19] for related problems. 2. Main results Our main result is as follows. Theorem 2.1. Let k≥3be an integer, qa prime number, aan integer with (a,q)=1and suppose Mand Nare integers with Nsatisfying N≤q1/2+1/(k1/2+1). (2.1)Then we have ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqo(1). We first note that the bound of Theorem 2.1 is non-trivial in the range q(1+ε)/(k−2)1/2≤N, (2.2) and, in this case, we have a bound of the form ∣∑M<n≤M+Neq(ank)∣≤N1−ρ+o(1), where ρ=2ε(1+ε)k(k+1). Comparing Theorem 2.1 with the bound (1.3), we see that Theorem 2.1 provides an improvement over (1.3) in the range N≥q2(k−1)/(k−3)(k−2)1/2=q2k−1/2+O(k−3/2). 3. Preliminary results We first prove the following general inequality for systems of congruences. Lemma 3.1. Suppose qis prime and t,mand ℓare integers. Let X⊆(Z/qZ)t,be a set and f={fi}i=1ma sequence of functions on X, fi:X→Z/qZ,i=1,…,m.Let σ={σi}i=12ℓ, be a sequence of numbers with each σi≢0modqand λ={λj}j=1ma sequence of numbers with each λj∈Z/qZ. We let Im,ℓ(f,X,σ,λ)denote the number of solutions to the system of congruences σ1fj(x1)+⋯+σ2ℓfj(x2ℓ)≡λjmodq,j=1,…,m, (3.1)with variables x1,…,x2ℓsatisfying x1,…,x2ℓ∈X.In the special case that σ={(−1)i}i=12ℓand each λj=0, we write Im,ℓ(f,X,{(−1)i}i=12ℓ,0)=Im,ℓ(f,X).For any X,f,σ and λas above, we have Im,ℓ(f,X,σ,λ)≤Im,ℓ(f,X). Proof Expanding the system (3.1) into additive characters, we see that Im,ℓ(f,X,σ,λ)=1qm∑1≤ah≤q1≤h≤m∏1≤i≤2ℓ(∑xi∈Xeq(σi(∑j=1majfj(xi))))eq(−∑j=1majλj), so that writing f(a1,…,am)=∑x∈Xeq(∑j=1majfj(x)), the above implies Im,ℓ(f,X,σ,λ)=1qm∑1≤ah≤q1≤h≤m∏1≤i≤2ℓf(σia1,…,σiam)eq(−∑j=1majλj), which we may bound by Im,ℓ(f,X,σ,λ)≤1qm∑1≤ah≤q1≤h≤m∏1≤i≤2ℓ∣f(σia1,…,σiam)∣. An application of Hölder’s inequality gives Im,ℓ(f,X,σ,λ)≤∏i=12ℓ(1qm∑1≤ah≤q1≤h≤m∣f(σia1,…,σiam)∣2ℓ)1/2ℓ, and hence Im,ℓ(f,X,σ,λ)≤∏i=12ℓ(1qm∑1≤ah≤q1≤h≤m∣f(a1,…,am)∣2ℓ)1/2ℓ=1qm∑1≤ah≤q1≤h≤m∣f(a1,…,am)∣2ℓ. The result follows since the term 1qm∑1≤ah≤q1≤h≤m∣f(a1,…,am)∣2ℓ, counts the number of solutions to the system of congruences ∑i=12ℓ(−1)ifj(xi)≡0,j=1,…,m, with variables x1,…,x2ℓ satisfying x1,…,x2ℓ∈X. □ The proof of the following uses the bound (1.7). Lemma 3.2. For integers k,r,Vand q, we let Jr,k(V;q)count the number of solutions to the system of congruences v1j+⋯+vrj≡vr+1j+⋯+v2rjmodq,j=1,…,k, (3.2)with variables satisfying 1≤v1,…,v2r≤V.Let 1≤m<kbe an integer and suppose Vsatisfies q1/m≪V<q1/mr. (3.3)If r≥k(k+1)/2, we have Jr,k(V;q)≤V2r−km+m(m−1)/2+o(1). Proof For integers λm+1,…,λr, we let Jr,k(V,λm+1,…,λk) denote the number of solutions to the system of equations v1j+⋯+vrj=vr+1j+⋯+v2rj,j=1,…,m, v1j+⋯+vrj=vr+1j+⋯+v2rj+λj,j=m+1,…,k, with variables v1,…,v2r satisfying 1≤v1,…,v2r≤V. Expressing Jr,k(V,λm+1,…,λk) via additive characters, we see that Jr,k(V,λm+1,…,λk)=∫01⋯∫01∣∑1≤v≤Ve(α1v+⋯+αkvk)∣2r×e(αm+1λm+1+⋯+αkλk)dα1…dαk, where e(z)=e2πiz, and hence Jr,k(V,λm+1,…,λk)≤∫01⋯∫01∣∑1≤v≤Ve(α1v+⋯+αkvk)∣2rdα1…dαk. This implies that Jr,k(V,λm+1,…,λk)≤Jr,k(V). (3.4) Using the assumption (3.3), we have Jr,k(V;q)=∑∣λj∣≤rVj/qm+1≤j≤kJr,k(V,λm+1q,…,λkq), and hence by (3.4) Jr,k(V;q)≪(∏j=m+1kVjq)Jr,k(V). Since r≥k(k+1)/2, an application of (1.7) gives Jr,k(V;q)≪(∏j=m+1kVjq)V2r−k(k+1)/2+o(1), which on recalling (3.3) simplifies to Jr,k(V;q)≪V2r−km+m(m−1)/2+o(1). □ For a proof of the following, see [10]. See also [2, 11, 15] for related and more precise results. Lemma 3.3. Let qbe prime and suppose M,Nand Uare integers with Nand Usatisfying NU≤q,U≤N. The number of solutions to the congruence n1u1≡n2u2modq, with variables satisfying M<n1,n2≤M+N,1≤u1,u2≤U, is bounded by O(NUlogq). Lemma 3.4. Let qbe prime and M,Nand Uintegers with Nand Usatisfying NU≤q,U≤N.Let ℓ≥1be an integer and suppose kis an integer satisfying 2ℓ≤k.Let Ik,ℓ(N,U)denote the number of solutions to the system of congruences n1ju1k−j+⋯+nℓjuℓk−j≡nℓ+1juℓ+1k−j+⋯+n2ℓju2ℓk−jmodq,j=0,…,2ℓ−1, (3.5)in variables n1,…,n2ℓ,u1,…,u2ℓsatisfying M<n1,…,n2ℓ≤M+N,1≤u1,…,u2ℓ≤U. (3.6)Then we have Ik,ℓ(N,U)≪(NU)ℓ(logq)ℓ−1. Proof We fix an integer k and proceed by induction on ℓ. Considering first the case ℓ=1, we recall that Ik,1(N,U) counts the number of solutions to the congruences u1k≡u2kmodqandu1k−1n1≡u2k−1n2modq, in variables M≤n1,n2≤M+N,1≤u1,u2≤U. Fixing a value of u1 determines u2 with at most k choices and fixing a value of n1 determines n2 with at most 1 choice. This implies that Ik,1(N,U)≪NU. We next assume that for integer ℓ satisfying ℓ−1≥1 and 2ℓ≤k, we have Ik,ℓ−1(N,U)≪(NU)ℓ−1(logq)ℓ−2, (3.7) and we aim to show that Ik,ℓ(N,U)≪(NU)ℓ(logq)ℓ−1. (3.8) Suppose the points n=(n1,…,n2ℓ) and u=(u1,…,u2ℓ) are a solution to (3.5) satisfying (3.6). Let H(u)⊆Fq2ℓ, denote the hyperplane H(u)={(z1,…,z2ℓ)∈Fq2ℓ:z1u1k+⋯−z2ℓu2ℓk≡0modq}. Considering the system (3.5), we see that the 2ℓ points pj(n,u)=((n1u1)j,…,(n2ℓu2ℓ)j),j=0,…,2ℓ−1, all lie on H(u). Since H(u) has dimension 2ℓ−1, this implies that there exists some non-trivial linear relation among the pj(n,u). In particular, there exists α0,…,α2ℓ−1∈Fq, with at least one αi≢0modq such that α0p0(n,u)+⋯+α2ℓ−1p2ℓ−1(n,u)≡0modq. (3.9) Equation (3.9) implies that the Vandermonde matrix of the points n1u1,…,n2ℓu2ℓ, is singular mod q, and hence there exist integers 1≤r,s≤2ℓ with r≠s such that nrur≡nsusmodq. (3.10) Letting Ik,ℓ(N,U,r,s) denote the number of solutions to the system (3.5) with variables satisfying (3.6) subject to the further condition (3.10), since the pair r,s can take at most 4ℓ2 values, we see that Ik,ℓ(N,U)≪Ik,ℓ(N,U,r,s), (3.11) for some pair r,s with r≠s. Considering Ik,ℓ(N,U,r,s), there exists some sequence of numbers σi∈{−1,1} for 1≤i≤2ℓ such that Ik,ℓ(N,U,r,s) is equal to the number of solutions to the system of congruences ∑1≤i≤2ℓi≠r,sσinijuik−j≡σrnrjurk−j+σsnsjusk−jmodq,j=0,…,2ℓ−1, (3.12) with variables satisfying (3.6) and (3.10). For each fixed nr,ns,ur,us, we let I0(nr,ns,ur,us) count the number of solutions to the system (3.12) with variables satisfying M<ni≤M+N,1≤ui≤U,i=1,…,2ℓ,i≠r,s, (3.13) so that the above implies Ik,ℓ(N,U)≪∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodqI0(nr,ns,ur,us). (3.14) Letting I0′ denote the number of solutions to the system of congruences n1ju1k−j+⋯+nℓ−1juℓ−1k−j≡nℓjuℓk−j+⋯+n2ℓ−2ju2ℓ−2k−jmodq,j=0,…,2ℓ−1, (3.15) in variables n1,…,n2ℓ−2,u1,…,u2ℓ−2 satisfying M<n1,…,n2ℓ−2≤M+N,1≤u1,…,u2ℓ−2≤U, (3.16) an application of Lemma 3.1 gives I0(nr,ns,ur,us)≤I0′, (3.17) for any nr,ns,ur and us. Considering only equations corresponding to j=0,…,2(ℓ−1)−1, in the system (3.15), we see that I0′≤Ik,ℓ−1(N,U), and hence by (3.17) I0(nr,ns,ur,us)≪Ik,ℓ−1(N,U). Combining the above with (3.14), we get Ik,ℓ(N,U)≪(∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodq1)Ik,ℓ−1(N,U). Since the term ∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodq1, counts the number of solutions to the congruence nrus≡nsurmodq, in variables nr,ns,ur,us satisfying M<nr,ns≤M+N,1≤ur,us≤U, an application of Lemma 3.3 gives ∑M<nr,ns≤M+N1≤ur,us≤Unrus≡nsurmodq1≪NUlogq, and hence Ik,ℓ(N,U)≪Ik,ℓ−1(N,U)NUlogq. Combining the above with (3.7), we see that Ik,ℓ(N,U)≪(NU)ℓ(logq)ℓ−1, which establishes (3.8).□ 4. Proof of Theorem 2.1 We proceed by induction on N and fix an arbitrarily small ε. Since the bound of Theorem 2.1 is trivial provided N≤q1/(k−2)1/2, this forms the basis of the induction. We formulate our induction hypothesis as follows. Let M be an arbitrary integer and suppose for all integers K≤N−1 we have ∣∑M<n≤M+Keq(ank)∣≤(q1/(k−2)1/2K)2/k(k+1)Kqε, uniformly over M. Using the above hypothesis, we aim to show ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqε. (4.1) We define the integers ℓ,m and r by ℓ=⌊k2⌋, (4.2) m=⌈2ℓ⌉, (4.3) r=k(k+1)2, (4.4) and define the integers U and V by U=⌊rN2q1/m⌋,V=⌊q1/m2r⌋. (4.5) We first note that UV≤N4. (4.6) Let 1≤u≤U and 1≤v≤V be integers and write ∑M<n≤M+Neq(ank)=∑M−uv<n≤M+N−uveq(a(n+uv)k)=∑M<n≤M+Neq(a(n+uv)k)+∑M−uv<n≤Meq(a(n+uv)k)−∑M+N−uv<n≤M+Neq(a(n+uv)k). Averaging the above over 1≤u≤U and 1≤v≤V, using (4.6) and applying our induction hypothesis, we see that ∣∑M<n≤M+Neq(ank)∣≤∣W∣UV+12(q1/(k−2)1/2N)2/k(k+1)Nqε, (4.7) where W=∑1≤u≤U∑1≤v≤V∑M<n≤M+Neq(a(n+uv)k). We have ∣W∣≤∑1≤v≤V∣∑1≤u≤U∑M<n≤M+Neq(a∑j=0k(kj)njuk−jvk−j)∣, hence by Hölder’s inequality ∣W∣ℓ≤Vℓ−1∑1≤v≤V∣∑1≤u≤U∑M<n≤M+Neq(a∑j=0k(kj)njuk−jvk−j)∣ℓ, so that for some complex numbers θv with ∣θv∣=1, we have ∣W∣ℓ≤Vℓ−1∑1≤ui≤U1≤i≤ℓ∑M<ni≤M+N1≤i≤ℓ∣∑1≤v≤Vθveq(a∑j=0k−1(kj)(n1ju1k−j+⋯+nℓjuℓk−j)vk−j)∣. Let W0=∑1≤ui≤U∑M<ni≤M+N∣∑1≤v≤Vθveq(a∑j=0k−1(kj)(n1ju1k−j+⋯+nℓjuℓk−j)vk−j)∣, so that ∣W∣ℓ≪Vℓ−1W0. (4.8) Let I(λ0,…,λk−1) denote the number of solutions to the system of congruences (kj)(n1ju1k−j+⋯+nℓjuℓk−j)≡λjmodq,j=0,…,k−1, in variables n1,…,nℓ,u1,…,uℓ satisfying M<n1,…,nℓ≤M+N,1≤u1,…,uℓ≤U. The above implies that W0=∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)∣∑1≤v≤Vθveq(a∑j=0k−1λjvk−j)∣. Two applications of Hölder’s inequality gives W02r≤(∑λ0,…,λk−1=0q−1I(λ0,…,λk−1))2r−2(∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2)×(∑λ0,…,λk−1=0q−1∣∑1≤v≤Vθveq(∑j=0k−1λjvk−j∣)2r). (4.9) The term ∑λ0,…,λk−1=0q−1∣∑1≤v≤Vθveq(∑j=0k−1λjvk−j)∣2r, is bounded by qk times the number of solutions to the system of congruences v1j+⋯+vrj≡vr+1j+⋯+v2rjmodq,j=1,…,k, in variables v1,…,v2r satisfying 1≤v1,…,v2r≤V. Recalling the choice of V in (4.5) and applying Lemma 3.2, we see that ∑λ0,…,λk−1=0q−1∣∑1≤v≤Vθveq(∑j=0k−1λjvk−j)∣2r≤qkJr,k(V;q)≤qkV2r−mk+m(m−1)/2+o(1). (4.10) We have ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)=(NU)ℓ, (4.11) and the term ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2, is equal to the number of solutions to the system of congruences n1ju1k−j+⋯+nℓjuℓk−j≡nℓ+1juℓ+1k−j+⋯+n2ℓju2ℓk−jmodq,j=0,…,k−1, (4.12) in variables n1,…,n2ℓ,u1,…,u2ℓ satisfying M<n1,…,n2ℓ≤M+N,1≤u1,…,u2ℓ≤U. Recalling (4.2) and considering only equations corresponding to j=0,…,2ℓ−1 in (4.12), we see that ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2≤Ik,ℓ(N,U), where Ik,ℓ(N,U) is defined as in Lemma 3.4. Recalling (2.2), we have logq≪logN=No(1), and hence ∑λ0,…,λk−1=0q−1I(λ0,…,λk−1)2≤Ik,ℓ(N,U)≤(NU)ℓ+o(1). (4.13) Combining (4.9) (4.10), (4.11) and (4.13) gives W02r≤qk+o(1)(NU)ℓ(2r−1)V2r−mk+m(m−1)/2, and hence by (4.8) ∣W∣ℓ≪qk/2r+o(1)Vℓ−mk/2r+m(m−1)/4r(NU)ℓ(1−1/2r). By (4.7) we have ∣∑M<n≤M+Neq(ank)∣≤(qkVmk−m(m−1)/2)1/2rℓN1−1/2rqo(1)U1/2r+12(q1/(k−2)1/2N)2/k(k+1)Nqε, which on recalling the choice of U and V in (4.5), the above simplifies to ∣∑M<n≤M+Neq(ank)∣≤q((m−1)/2ℓ+1/m)/2r+o(1)N1−1/r+12(q1/(k−2)1/2N)2/k(k+1)Nqε. Recalling the choice of ℓ, m and r in (4.2), (4.3) and (4.4), we get ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqo(1)+12(q1/(k−2)1/2N)2/k(k+1)Nqε, and hence ∣∑M<n≤M+Neq(ank)∣≤(q1/(k−2)1/2N)2/k(k+1)Nqε, by taking the term o(1) in qo(1) to be sufficiently small. Acknowledgement The author would like to thank Igor Shparlinski for bringing the paper of Karatsuba [14] to the author’s attention, for explaining the Russian version of Karatsuba’s paper and for his comments on a preliminary version of the current paper. References 1 G. I. Arkhipov and A. A. 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Wooley , The cubic case of the main conjecture in Vinogradov’s mean value theorem , Adv. Math. 294 ( 2016 ), 532 – 561 . Google Scholar CrossRef Search ADS 28 T. D. Wooley , Nested efficient congruencing and relatives of Vinogradov’s mean value theorem, arXiv:1708.01220. © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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