Point-Line Incidence Estimates in $(\mathbb{Z}/p^k\mathbb{Z})^2$

TL;DR

This paper extends incidence bounds from finite fields to the $p$-adic setting, providing new bounds for points and lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$, including cases with non-separated lines.

Contribution

It introduces the first incidence bounds in the $p$-adic setting, generalizing finite field results and handling non-separated lines with Fourier analysis and induction-on-scales.

Findings

Extended incidence bounds to $(\mathbb{Z}/p^k\mathbb{Z})^2$

Established bounds for non-separated lines

Applied Fourier analysis and induction-on-scales techniques

Abstract

The point-line incidence problem has been widely studied in Euclidean spaces and vector spaces over finite fields, whereas the analogous problem has rarely been considered over finite $p$-adic rings. In this paper, we investigate incidences in the $p$-adic setting and prove new incidence bounds for points and lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$. Our first two results extend previously known incidence bounds over finite fields, assuming lines are well-separated. For non-separated lines, we establish a general incidence result for weighted points and lines under certain dimensional spacing conditions using the Fourier analytic method and the induction-on-scales argument.