Bounds for discrete moments of Weyl sums and applications

Abstract

We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The proofs both build on the recently proved main conjecture for Vinogradov's mean value theorem.We present two selected applications: First, we prove a new kth derivative test for the number of integer points close to a curve by an exponential sum approach. This yields a stronger bound than existing results obtained via geometric methods, but it is only applicable for specific functions. As second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree k ≥ 4.1991 Mathematics Subject Classification. Primary 11L15, Secondary 11J54.

Large sieve estimate for multivariate polynomial moduli and applications

“…This was proved by the second author as [22,Thm. 1.2] with k + 1 instead of k − 1, thus confirming [15,Conjecture 21] with ω = 1/k(k + 1). Some improvements over the bound (2.6) have been obtained by the first author using Weyl sums estimates and the recent breakthroughs in Vinogradov's mean value theorem from [6,28] and its generalizations to general polynomials of several variables [23].…”

“…), but we are still far away from proving this conjecture even for polynomials in one variable. It was conjectured in [15] that for the one-dimensional case ℓ = 1, one should be able to reach 1) . This was proved by the second author as [22,Thm.…”

Weyl sums over integers with digital restrictions

“…Remark 3.4. We note that [16,Theorem 9] implies a slightly weaker version of (3.4). Our improvement is due to the fact that we enter the details of the proof of [30,Theorem 5.2].…”

Large sieve inequality for power moduli

“…The range where it improves all the previous bounds becomes non empty as soon as k ≥ 4 and covers almost the whole range except the corners. Additionally it is a step towards Conjecture 21 raised in [5].…”