Distribution of distances in positive characteristic

2019/05/16 by Pham, Thang, Vinh, Le Anh
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1905.06483

Abstract

Let \mathbbF_q be an arbitrary finite field, and E be a set of points in \mathbbF_q^d. Let Δ(E) be the set of distances determined by pairs of points in E. By using the Kloosterman sums, Iosevich and Rudnev proved that if |E|≥ 4q^(d+1)/(2), then Δ(E)=\mathbbF_q. In general, this result is sharp in odd-dimensional spaces over arbitrary finite fields. In this paper, we use the recent point-plane incidence bound due to Rudnev to prove that if E has Cartesian product structure in vector spaces over prime fields, then we can break the exponent (d+1)/2, and still cover all distances. We also show that the number of pairs of points in E of any given distance is close to its expected value.

Related

• Structural theorems on the distance sets over finite fields
• On the structure of distance sets over prime fields
• Pinned distance sets, k-simplices, Wolff's exponent in finite fields and sum-product estimates
• Pinned distance sets, Wolff's exponent in finite fields and improved sum-product estimates
• Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture