On the structure of distance sets over prime fields

2018/12/30 by Pham, Thang, Suk, Andrew · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1812.11556

Abstract

Let \mathbbF_q be a finite field of order q and E be a set in \mathbbF_q^d. The distance set of E, denoted by Δ(E), is the set of distinct distances determined by the pairs of points in E. Very recently, Iosevich, Koh, and Parshall (2018) proved that if |E|≫ q^d/2, then the quotient set of Δ(E) satisfies \vert(Δ(E))/(Δ(E))\vert=\vert \lbrace(a)/(b)\colon a, b∈ Δ(E), b≠ 0\rbrace\vert≫ q. In this paper, we break the exponent d/2 when E is a Cartesian product of sets over a prime field. More precisely, let p be a prime and A⊂ \mathbbF_p. If E=A^d⊂ \mathbbF_p^d and |E|≫ p^(d)/(2)-ε for some ε>0, then we have \vert(Δ(E))/(Δ(E))\vert, ~\vert Δ(E)⋅ Δ(E)\vert ≫ p. Such improvements are not possible over arbitrary finite fields. These results give us a better understanding about the structure of distance sets and the Erdős-Falconer distance conjecture over finite fields.

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• Erdős--Falconer distance conjecture from an analytic perspective

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