The distribution of values of short hybrid exponential sums on curves over finite fields

TL;DR

This paper investigates the distribution of hybrid exponential sums on algebraic curves over finite fields, demonstrating that their projections tend to a Gaussian distribution as the field size increases.

Contribution

It establishes the Gaussian limiting distribution of the sums' projections on curves over finite fields, under natural conditions.

Findings

Projections of sums follow a Gaussian distribution as p increases.

Distribution results hold for sums over short intervals.

Provides conditions under which the Gaussian behavior emerges.

Abstract

Let $p$ be a prime number, $X$ be an absolutely irreducible affine plane curve over $\mathbb{F}_p$, and $g,f\in\mathbb{F}_p(x,y)$. We study the distribution of the values of the hybrid exponential sums S_n on $n\in\mathcal{I}$ for some short interval $\mathcal{I}$. We show that under some natural conditions the limiting distribution of the projections of the sum $S_n$, $n\in\mathcal{I}$ on any straight line through the origin is Gaussian as $p$ tends to infinity.