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On the structure of distance sets over prime fields


Authors: Thang Pham and Andrew Suk
Journal: Proc. Amer. Math. Soc. 148 (2020), 3209-3215
MSC (2010): Primary 11T06, 11T60, 11B30
DOI: https://doi.org/10.1090/proc/15052
Published electronically: April 22, 2020
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Abstract: Let $ \mathbb{F}_q$ be a finite field of order $ q$, and let $ \mathcal {E}$ be a set in $ \mathbb{F}_q^d$. The distance set of $ \mathcal {E}$, denoted by $ \Delta (\mathcal {E})$, is the set of distinct distances determined by the pairs of points in $ \mathcal {E}$. Very recently, Iosevich, Koh, and Parshall (2018) proved that if $ \vert\mathcal {E}\vert\gg q^{d/2}$, then the quotient set of $ \Delta (\mathcal {E})$ satisfies

$\displaystyle \left \vert \frac {\Delta (\mathcal {E})}{\Delta (\mathcal {E})}\... ...}\colon a, b\in \Delta (\mathcal {E}), b\ne 0\right \rbrace \right \vert \gg q.$

In this paper, we break the exponent $ d/2$ when $ \mathcal {E}$ is a Cartesian product of sets over a prime field. More precisely, let $ p$ be a prime, and let $ A\subset \mathbb{F}_p$. If $ \mathcal {E}=A^d\subset \mathbb{F}_p^d$ and $ \vert\mathcal {E}\vert\gg p^{\frac {d}{2}-\varepsilon }$ for some $ \varepsilon >0$, then we have

$\displaystyle \left \vert \frac {\Delta (\mathcal {E})}{\Delta (\mathcal {E})}\... ...\left \vert \Delta (\mathcal {E})\cdot \Delta (\mathcal {E})\right \vert \gg p.$

Such improvements are not possible over arbitrary finite fields. These results give us a better understanding about the structure of distance sets and the Erdős-Falconer distance conjecture over finite fields.

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Additional Information

Thang Pham
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email: v9pham@ucsd.edu

Andrew Suk
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email: asuk@ucsd.edu

DOI: https://doi.org/10.1090/proc/15052
Received by editor(s): December 30, 2018
Published electronically: April 22, 2020
Additional Notes: The first author was supported by Swiss National Science Foundation grant P2ELP2-175050.
The second author was supported by an NSF CAREER award and an Alfred Sloan Fellowship.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2020 American Mathematical Society