Incidences of M\"obius transformations in $\mathbb F_p$

TL;DR

This paper extends incidence theorems involving Möbius transformations over finite fields, providing new bounds and applications in additive combinatorics and discrete geometry.

Contribution

It develops methods to prove incidence bounds between Möbius transformations and point sets in finite fields, including asymmetric cases and applications.

Findings

Established new incidence bounds for Möbius transformations in finite fields

Derived asymmetric incidence results with potential applications

Applied incidence theorems to problems in additive combinatorics and geometry

Abstract

We develop the methods used by Rudnev and Wheeler to prove an incidence theorem between arbitrary sets of M\"{o}bius transformations and point sets in $\mathbb F_p^2$. We also note some asymmetric incidence results, and give applications of these results to various problems in additive combinatorics and discrete geometry.