Vol. 8, No. 2, 2019
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On the quotient set of the distance set

Alex Iosevich, Doowon Koh and Hans Parshall
Vol. 8 (2019), No. 2, 103–115
DOI: 10.2140/moscow.2019.8.103
Abstract

Let Fq be a finite field of order q. We prove that if d ≥ 2 is even and E ⊂ Fqd with |E|≥ 9qd∕2 then

Fq = Δ(E) Δ(E) ={a b : a ∈ Δ(E),b ∈ Δ(E)∖{0}},

where

Δ(E) = {∥x − y∥ : x,y ∈ E},∥x∥ = x12 + x 22 + ⋯ + x d2.

If the dimension d is odd and E ⊂ Fqd with |E|≥ 6qd∕2, then

{0}∪ Fq+ ⊂ Δ(E) Δ(E),

where Fq+ denotes the set of nonzero quadratic residues in Fq. Both results are, in general, best possible, including the conclusion about the nonzero quadratic residues in odd dimensions.

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Keywords
quotient set, distance set, finite field
Mathematical Subject Classification 2010
Primary: 11T24, 52C17
Milestones
Received: 5 March 2018
Revised: 24 November 2018
Accepted: 15 December 2018
Published: 20 May 2019
Authors
Alex Iosevich
Department of Mathematics
University of Rochester
Rochester, NY
United States
Doowon Koh
Department of Mathematics
Chungbuk National University
Cheongju
South Korea
Hans Parshall
Department of Mathematics
The Ohio State University
Columbus, OH
United States