An asymmetric bound for sum of distance sets

2020/08/19 by Cheong, Daewoong, Koh, Doowon, Pham, Thang · 1 citation
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2008.08344

Abstract

For E⊂ \mathbbF_q^d, let Δ(E) denote the distance set determined by pairs of points in E. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if E,F⊂ \mathbbF_q^d are subsets with |E||F|≫ q^d+(1)/(3) then |Δ(E)+Δ(F)|> q/2. They also proved that the threshold q^d+(1)/(3) is sharp when |E|=|F|. In this paper, we provide an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in our proofs is an optimal L² restriction theorem for the sphere of zero radius.

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• The Erdős-Falconer distance problem between arbitrary sets and k-coordinatable sets in finite fields

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