Moscow Journal of Combinatorics and Number Theory

On the quotient set of the distance set

Alex Iosevich, Doowon Koh, and Hans Parshall

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Abstract

Let Fq be a finite field of order q. We prove that if d2 is even and EFqd with |E|9qd2 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 EFqd with |E|6qd2, 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

Subjects
Primary: 11T24: Other character sums and Gauss sums 52C17: Packing and covering in $n$ dimensions [See also 05B40, 11H31]

Keywords
quotient set distance set finite field

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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