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

 Download this article For screen For printing  Recent Issues Volume 11, Issue 4 Volume 11, Issue 3 Volume 11, Issue 2 Volume 11, Issue 1 Volume 10, Issue 4 Volume 10, Issue 3 Volume 10, Issue 2 Volume 10, Issue 1 Volume 9, Issue 4 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  Older Issues Volume 7, Issue 4 Volume 7, Issue 3 Volume 7, Issue 2 Volume 7, Issue 1 Volume 6, Issue 4 Volume 6, Issue 2-3 Volume 6, Issue 1 Volume 5, Issue 4 Volume 5, Issue 3 Volume 5, Issue 1-2 Volume 4, Issue 4 Volume 4, Issue 3 Volume 4, Issue 2 Volume 4, Issue 1 Volume 3, Issue 3-4 Volume 3, Issue 2 Volume 3, Issue 1 Volume 2, Issue 4 Volume 2, Issue 3 Volume 2, Issue 2 Volume 2, Issue 1 Volume 1, Issue 4 Volume 1, Issue 3 Volume 1, Issue 2 Volume 1, Issue 1  The Journal About the Journal Editorial Board Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement founded and published with the scientific support and advice of mathematicians from the Moscow Institute of Physics and Technology ISSN (electronic): 2640-7361 ISSN (print): 2220-5438 Author Index To Appear 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.

We have not been able to recognize your IP address 3.239.119.61 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form. 