The distribution of rational points and polynomial maps on an affine variety over a finite field on average

TL;DR

This paper investigates the distribution of rational points on an affine variety over finite fields, establishing conditions under which most translated regions contain the expected number of points, thus extending Weil estimates to smaller regions.

Contribution

It provides a lower bound on the volume of regions ensuring most translations contain the expected number of points, demonstrating Weil estimates in smaller regions on average.

Findings

Almost all translations of sufficiently large boxes contain the expected number of points.

Weil estimates hold in smaller regions for most translations.

The results apply to absolutely irreducible affine varieties over finite fields.

Abstract

Let $p$ be a prime, let $V/\mathbb{F}_p$ be an absolutely irreducible affine variety inside the affine $r$-space. In this paper, we consider the problem of how often a box $B$ will contain the expected number of points. In particular, we give a lower bound on the volume of $B$ that guarantees almost all translations of $B$ in the $r$-space contain the expected number of points. This shows that the Weil estimate holds in smaller regions in an "almost all" sense.