On character sums over flat numbers

TL;DR

This paper investigates bounds for character sums over a special set of integers called flat numbers, which are close to their modular inverses, extending understanding of character sums in number theory.

Contribution

The authors establish new estimates for character sums over flat numbers, a novel set defined by proximity to modular inverses, advancing the analysis of character sums in modular arithmetic.

Findings

Derived bounds for character sums over flat numbers

Extended previous results to a broader class of sums

Provided new techniques for estimating sums involving modular inverses

Abstract

Let $q\geqslant2$ be an integer, $\chi$ be any non-principal character mod $q$, and $H=H(q)\leqslant q.$ In this paper the authors prove some estimates for character sums of the form \[\mathcal{W}(\chi,H;q)=\sum_{n\in\mathscr{F}(H)}\chi(n),\] where \[\mathscr{F}(H)=\left\{n\in\mathbb{Z}|(n,q)=1,1\leqslant n,\bar{n}\leqslant q, |n-\bar{n}|\leqslant H\},\] $\bar{n}$ is defined by $n\bar{n}\equiv1\pmod q.$