Lehmer points and visible points on affine varieties over finite fields

TL;DR

This paper studies the distribution and count of special points on affine varieties over finite fields, focusing on Lehmer points with congruence conditions and visible points with coprime coordinates, providing asymptotic formulas.

Contribution

It introduces new asymptotic results for counting Lehmer and visible points on affine varieties over finite fields and analyzes their distribution across congruence classes.

Findings

Asymptotic formulas for the number of Lehmer points

Asymptotic formulas for the number of visible points

Distribution analysis of visible points in different classes

Abstract

Let $V$ be an absolutely irreducible affine variety over $\mathbb{F}_p$. A Lehmer point on $V$ is a point whose coordinates satisfy some prescribed congruence conditions, and a visible point is one whose coordinates are relatively prime. Asymptotic results for the number of Lehmer points and visible points on $V$ are obtained, and the distribution of visible points into different congruence classes is investigated.