Vol. 8, No. 2, 2019

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On the quotient set of the distance set

Alex Iosevich, Doowon Koh and Hans Parshall

Vol. 8 (2019), No. 2, 103–115
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

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

where

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

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

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

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.

Keywords
quotient set, distance set, finite field
Mathematical Subject Classification 2010
Primary: 11T24, 52C17