New bounds for distance-type problems over prime fields
type: Item
instance of: scholarly article
title: New bounds for distance-type problems over prime fields
zbMATH Open document ID: 1478.11031
author: Alex Iosevich
author: Doowon Koh
author: Pham Van Thang
published in: European Journal of Combinatorics
publication date: 2020-04-09T00:00:00Z
review text: This paper is concerned with the prime field analogue of the Erdős distinct distances problem. Given a point set \(P \subset \mathbb F_p^2\), define \[ D(P):= \{ \| x-y \| : x,y \in P \}, \] where \(x=(x_1,x_2)\), \(y=(y_1,y_2)\) and \[ \| x-y \|= (x_1-y_1)^2+(x_2-y_2)^2. \] A simple argument shows that the bound \[ |D(P)| \gg |P|^{1/2} \] holds for any \(P \subset \mathbb F_p^2\). The exponent was improved to \(1/2 +c\) by [\textit{J. Bourgain} et al., Geom. Funct. Anal. 14, No. 1, 27--57 (2004; Zbl 1145.11306)], under the necessary constraint that the point set is not too large. Such a condition is necessary to avoid degenerate situations such as the case when \(P=\mathbb F_p^2\), in which case \(|D(P)|=p=|P|^{1/2}\). Recent developments in finite field incidence theory, particularly Rudnev's point-plane incidence theorem [\textit{M. Rudnev}, Combinatorica 38, No. 1, 219--254 (2018; Zbl 1413.51001)], have led to quantitative progress for this problem. \textit{S. Stevens} and \textit{F. de Zeeuw} [Bull. Lond. Math. Soc. 49, No. 5, 842--858 (2017; Zbl 1388.51002)] obtained the bound \(|D(P)| \gg |P|^{8/15}\). There have since been improvements to this result, and this paper gives a further improvement. The main result of the paper is that the bound \[ |D(P)| \gg |P|^{\frac{1}{2} + \frac{69}{1558} -o(1)}\tag{1} \] holds for all sufficiently small \(P \subset \mathbb F_p^2\), provided that \(p\) is congruent to \(3\) modulo \(4\). Such a condition is necessary because of issues arising from the existence of isotropic lines when \(p \equiv 1 \mod 4\). Note that a subsequent paper of \textit{B. Murphy} et al. [``On the pinned distances problem over finite fields'', Preprint, \url{arXiv:2003.00510}] gives a further improvement, establishing the bound \[ |D(P)| \gg |P|^{2/3}\tag{2} \] provided that \(P\) is sufficiently small and not contained in a single isotropic line. The proof of (2) establishes a more direct and efficient connection between point-plane incidences and…
reviewed by: Oliver Roche-Newton
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 52C10
zbMATH DE Number: 7188986
zbMATH Keywords: distances
zbMATH Keywords: finite fields
zbMATH Keywords: sum-product problems
MaRDI profile type: Publication
full work available at URL: j.ejc.2020.103080
OpenAlex ID: W3003942000
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DOI: 10.1016/J.EJC.2020.103080
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description: scientific article; zbMATH DE number 7188986