Distribution of differences of characters evaluated at consecutive polynomial values

TL;DR

This paper investigates the distribution of differences of additive and multiplicative characters evaluated at consecutive polynomial values over finite fields, deriving formulas for moments and analyzing their behavior under certain conditions.

Contribution

It introduces new methods to analyze the distribution of character differences at polynomial points, providing explicit formulas for moments and extending understanding of character sums.

Findings

Derived formulas for the first moment of character differences

Analyzed distribution properties of character sums at polynomial values

Extended results to specific conditions on characters and polynomials

Abstract

In this paper, we study the distribution of difference of multiplicative and additive characters modulo $p$ at consecutive polynomial values. More precisely, for an interval $I$ over finite field and $0<m<1$, we investigate the following sums \begin{align*} \sum_{n\in I}|\psi(F(n))-\psi(F(n+1))|^{2m} \quad \text{and} \quad \sum_{n\in I}|\chi(F(n))-\chi(F(n+1))|^{2m}, \end{align*} where $\psi$ is a non-trivial additive character and $\chi$ is a non-trivial multiplicative character modulo $p$, under suitable conditions on $\chi$ and $F$. As a consequence, we derive a formula for the first moment by specializing to $m=1/2$.