On Character Sums and Exponential Sums over Generalized Arithmetic Progressions

TL;DR

This paper establishes new uniform upper bounds for Dirichlet character sums over generalized arithmetic progressions, extending classical inequalities and applying Fourier analysis techniques.

Contribution

It introduces a generalized bound for character sums over proper generalized arithmetic progressions, broadening the scope of classical inequalities like Polya and Vinogradov.

Findings

Proves a uniform upper bound for character sums over generalized arithmetic progressions.

Extends classical inequalities to a broader class of sets.

Provides bounds for polynomial exponential sums using Fourier analysis.

Abstract

We study upper bounds for sums of Dirichlet characters. We prove a uniform upper bound of the character sum over all proper generalized arithmetic progressions, which generalizes the classical Polya and Vinogradov inequality. Our argument is based on getting an upper bound for the l1 norm of the Fourier coefficients of a generalized arithmetic progression. Our method also applies to give upper bounds for polynomial exponential sums.