#### Vol. 8, No. 2, 2019

 Download this article For screen For printing  Recent Issues Volume 9, Issue 3 Volume 9, Issue 2 Volume 9, Issue 1 Volume 8, Issue 4 Volume 8, Issue 3 Volume 8, Issue 2 Volume 8, Issue 1  The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN (electronic): 2640-7361 ISSN (print): 2220-5438 Previously Published To Appear founded and published with the scientific support and advice of the Moscow Institute of Physics and Technology Other MSP Journals  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.

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