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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 ${\mathbb{F}}_{q}$ be a finite field of order $q$. We prove that if $d\ge 2$ is even and $E\subset {\mathbb{F}}_{q}^{d}$ with $|E|\ge 9{q}^{d∕2}$ then

${\mathbb{F}}_{q}=\frac{\mathrm{\Delta }\left(E\right)}{\mathrm{\Delta }\left(E\right)}=\left\{\frac{a}{b}:a\in \mathrm{\Delta }\left(E\right),b\in \mathrm{\Delta }\left(E\right)\setminus \left\{0\right\}\right\},$

where

$\mathrm{\Delta }\left(E\right)=\left\{\parallel x-y\parallel :x,y\in E\right\},\phantom{\rule{1em}{0ex}}\parallel x\parallel ={x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{d}^{2}.$

If the dimension $d$ is odd and $E\subset {\mathbb{F}}_{q}^{d}$ with $|E|\ge 6{q}^{d∕2}$, then

$\left\{0\right\}\cup {\mathbb{F}}_{q}^{+}\subset \frac{\mathrm{\Delta }\left(E\right)}{\mathrm{\Delta }\left(E\right)},$

where ${\mathbb{F}}_{q}^{+}$ denotes the set of nonzero quadratic residues in ${\mathbb{F}}_{q}$. Both results are, in general, best possible, including the conclusion about the nonzero quadratic residues in odd dimensions.

Citation

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  