Weyl sums over integers with digital restrictions

TL;DR

This paper estimates Weyl sums over integers with binary digit restrictions, utilizing advanced number theory tools, and applies these results to analyze the discrepancy of certain polynomial-defined point sets.

Contribution

It introduces new bounds for Weyl sums with binary digit restrictions using recent advances in the Vinogradov mean value theorem.

Findings

Derived bounds for Weyl sums with binary digit restrictions

Applied results to estimate discrepancy of polynomial-based point sets

Utilized recent breakthroughs in Vinogradov mean value theorem

Abstract

We estimate Weyl sums over the integers with sum of binary digits either fixed or restricted by some congruence condition. In our proofs we use ideas that go back to a paper by Banks, Conflitti and the first author (2002). Moreover, we apply the "main conjecture" on the Vinogradov mean value theorem which has been established by Bourgain, Demeter and Guth (2016) as well as by Wooley (2016, 2019). We use our result to give an estimate of the discrepancy of point sets that are defined by the values of polynomials at arguments having the sum of binary digits restricted in different ways.