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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