## Moscow Journal of Combinatorics and Number Theory

### On the quotient set of the distance set

#### 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

$F q = Δ ( E ) Δ ( E ) = { a b : a ∈ Δ ( E ) , b ∈ Δ ( E ) ∖ { 0 } } ,$

where

$Δ ( E ) = { ∥ x − y ∥ : x , y ∈ E } , ∥ x ∥ = x 1 2 + x 2 2 + ⋯ + x d 2 .$

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

${ 0 } ∪ F q + ⊂ Δ ( E ) Δ ( E ) ,$

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.

#### Article information

Source
Mosc. J. Comb. Number Theory, Volume 8, Number 2 (2019), 103-115.

Dates
Received: 5 March 2018
Revised: 24 November 2018
Accepted: 15 December 2018
First available in Project Euclid: 29 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.moscow/1559095385

Digital Object Identifier
doi:10.2140/moscow.2019.8.103

Mathematical Reviews number (MathSciNet)
MR3959878

Zentralblatt MATH identifier
07063267

#### Citation

Iosevich, Alex; Koh, Doowon; Parshall, Hans. On the quotient set of the distance set. Mosc. J. Comb. Number Theory 8 (2019), no. 2, 103--115. doi:10.2140/moscow.2019.8.103. https://projecteuclid.org/euclid.moscow/1559095385

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