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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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 } } ,


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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

quotient set distance set finite field


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.

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