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arXiv · 2505.11859

Distribution of differences of characters evaluated at consecutive polynomial values

Nilanjan Bag·Dwaipayan Mazumder
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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}|ψ(F(n))-ψ(F(n+1))|^{2m} \quad \text{and} \quad \sum_{n\in I}|χ(F(n))-χ(F(n+1))|^{2m}, \end{align*} where $ψ$ is a non-trivial additive character and $χ$ is a non-trivial multiplicative character modulo $p$, under suitable conditions on $χ$ and $F$. As a consequence, we derive a formula for the first moment by specializing to $m=1/2$.

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Nilanjan Bag, Dwaipayan Mazumder. 2026-01-29. Distribution of differences of characters evaluated at consecutive polynomial values. https://arxiv.org/abs/2505.11859

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