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2019 On the quotient set of the distance set
Alex Iosevich, Doowon Koh, Hans Parshall
Mosc. J. Comb. Number Theory 8(2): 103-115 (2019). DOI: 10.2140/moscow.2019.8.103

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.

Citation

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Alex Iosevich. Doowon Koh. Hans Parshall. "On the quotient set of the distance set." Mosc. J. Comb. Number Theory 8 (2) 103 - 115, 2019. https://doi.org/10.2140/moscow.2019.8.103

Information

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

zbMATH: 07063267
MathSciNet: MR3959878
Digital Object Identifier: 10.2140/moscow.2019.8.103

Subjects:
Primary: 11T24, 52C17

Rights: Copyright © 2019 Mathematical Sciences Publishers

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