Distribution of distances in positive characteristic

Thang Pham and Lê Anh Vinh

Vol. 309 (2020), No. 2, 437–451

DOI: 10.2140/pjm.2020.309.437

Abstract

Let ${\mathbb{𝔽}}_q$ be an arbitrary finite field, and $\mathcal{ℰ}$ be a point set in ${\mathbb{𝔽}}_q^d$. Let $\Delta \left(\mathcal{ℰ}\right)$ be the set of distances determined by pairs of points in $\mathcal{ℰ}$. Using Kloosterman sums, Iosevich and Rudnev (2007) proved that if $\left|\mathcal{ℰ}\right|\ge 4q^{\left(d+1\right)∕2}$ then $\Delta \left(\mathcal{ℰ}\right)={\mathbb{𝔽}}_q$. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. We use the point-plane incidence bound due to Rudnev to prove that if $\mathcal{ℰ}$ has Cartesian product structure in vector spaces over prime fields, then we can break the exponent $\left(d+1\right)∕2$ and still cover all distances. We also show that the number of pairs of points in $\mathcal{ℰ}$ of any given distance is close to its expected value.

Keywords

distances, finite fields, incidence, Rudnev's point-plane incidence bound

Mathematical Subject Classification

Primary: 14N10, 51A45, 52C10

Milestones

Received: 18 June 2020

Revised: 22 July 2020

Accepted: 29 July 2020

Published: 14 January 2021