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ON THE TWO-PARAMETER ERDŐS–FALCONER DISTANCE PROBLEM IN FINITE FIELDS

Published online by Cambridge University Press:  29 September 2022

FRANCOIS CLÉMENT
Affiliation:
Sorbonne Université, CNRS, LIP6, Paris, France e-mail: francois.clement@lip6.fr
HOSSEIN NASSAJIAN MOJARRAD
Affiliation:
Courant Institute, New York University, New York 10012, USA e-mail: sn2854@nyu.edu
DUC HIEP PHAM*
Affiliation:
University of Education, Vietnam National University, Hanoi, Vietnam
CHUN-YEN SHEN
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan e-mail: cyshen@math.ntu.edu.tw

Abstract

Given $E \subseteq \mathbb {F}_q^d \times \mathbb {F}_q^d$ , with the finite field $\mathbb {F}_q$ of order q and the integer $d\,\ge \, 2$ , we define the two-parameter distance set $\Delta _{d, d}(E)=\{(\|x-y\|, \|z-t\|) : (x, z), (y, t) \in E \}$ . Birklbauer and Iosevich [‘A two-parameter finite field Erdős–Falconer distance problem’, Bull. Hellenic Math. Soc. 61 (2017), 21–30] proved that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}(E)| = q^2$ . For $d=2$ , they showed that if $|E| \gg q^{{10}/{3}}$ , then $ |\Delta _{2, 2}(E)| \gg q^2$ . In this paper, we give extensions and improvements of these results. Given the diagonal polynomial $P(x)=\sum _{i=1}^da_ix_i^s\in \mathbb F_q[x_1,\ldots , x_d]$ , the distance induced by P over $\mathbb {F}_q^d$ is $\|x-y\|_s:=P(x-y)$ , with the corresponding distance set $\Delta ^s_{d, d}(E)=\{(\|x-y\|_s, \|z-t\|_s) : (x, z), (y, t) \in E \}$ . We show that if $|E| \gg q^{{(3d+1)}/{2}}$ , then $ |\Delta _{d, d}^s(E)| \gg q^2$ . For $d=2$ and the Euclidean distance, we improve the former result over prime fields by showing that $ |\Delta _{2,2}(E)| \gg p^2$ for $|E| \gg p^{{13}/{4}}$ .

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The second-listed author was supported by Swiss National Science Foundation grant P2ELP2-178313.

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