Vol. 309, No. 2, 2020

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Distribution of distances in positive characteristic

Thang Pham and Lê Anh Vinh

Vol. 309 (2020), No. 2, 437–451
Abstract

Let 𝔽q be an arbitrary finite field, and be a point set in 𝔽qd. Let Δ() be the set of distances determined by pairs of points in . Using Kloosterman sums, Iosevich and Rudnev (2007) proved that if || 4q(d+1)2 then Δ() = 𝔽q. 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 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 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
Authors
Thang Pham
Department of Mathematics
ETH, Zurich
Zurich
Switzerland
Lê Anh Vinh
Vietnam Institute of Educational Sciences
Ha Noi
Vietnam