Incidence bounds with M\"obius hyperbolae in positive characteristic

TL;DR

This paper establishes improved incidence bounds between Cartesian product point sets and M"obius hyperbolae in fields of large positive characteristic, advancing understanding of geometric configurations and their combinatorial properties.

Contribution

It introduces a novel intermediate bound on k-rich hyperbolae and extends incidence bounds to more general sets, surpassing previous results by Shkredov.

Findings

Enhanced incidence bounds in positive characteristic fields.

Generalization from Cartesian products to broader sets.

Connection between energy estimates and the Erdős distance problem.

Abstract

We prove new incidence bounds between a plane point set, which is a Cartesian product, and a set of translates $H$ of the hyperbola $xy=\lambda\neq 0$, over a field of asymptotically large positive characteristic $p$. They improve recent bounds by Shkredov, which are based on using explicit incidence estimates in the early terminated procedure of repeated applications of the Cauchy-Schwarz inequality, underlying many qualitative results related to growth and expansion in groups. The improvement -- both quantitative, plus we are able to deal with a general $H$, rather than a Cartesian product -- is mostly due to a non-trivial "intermediate" bound on the number of $k$-rich M\"obius hyperbolae in positive characteristic. In addition, we make an observation that a certain energy-type quantity in the context of $H$ can be bounded via the $L^2$-moment of the Minkowski distance in $H$ and can therefore fetch the corresponding estimates apropos of the Erd\H{o}s distinct distance problem.