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

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.

submitted. Added some more corollaries

References in corpus (2)

• Points on curves in small boxes en applications
• The sliding-sum method for short exponential sums

Cited by in corpus (1)

• The distribution of points on curves over finite fields in some small rectangles