New bounds for distance-type problems over prime fields
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 the size of \(D(P)\). In the paper under review, the authors prove (1) by establishing and exploiting a connection between the size of \(D(P)\), the number of isosceles triangles contained in \(P\), and the number of rectangles contained in \(P\). The authors also use recent developments in finite field sum-product theory to prove the expander bound \[ |(A-A)^2-(A-A)^2| \gg |A|^{\frac{3}{2}+\frac{1}{142}-o(1)},\tag{3} \] provided that \(A\) is not too large. The set \((A-A)^2 -(A-A)^2\) is related by a change of sign to \(D(A \times A)=(A-A)^2+(A-A)^2\). \textit{G. Petridis} [Proc. Am. Math. Soc. 145, No. 11, 4639--4645 (2017; Zbl 1422.11017)] had previously proved the bound \[ |(A-A)^2 +(A-A)^2| \gg |A|^{\frac{3}{2}}. \] In order to prove (3), the authors prove a sum-product type bound concerning the realtionship between the sizes of \(|A-A|\) and \(|A^2-A^2|\) which breaks an interesting and difficult threshold.
scientific article; zbMATH DE number 7188986
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