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

TL;DR

This work investigates point-line incidences in the finite p-adic ring $(\mathbb{Z}/p^k\mathbb{Z})^2$. It extends Szemerédi–Trotter–type bounds to the $p$-adic setting for well-separated lines by reducing to finite-field configurations and applying projective-transform arguments, yielding incidence bounds that mirror the finite-field theory. For non-separated lines, the authors develop a generalized incidence framework with weighted points/lines and dimensional spacing conditions, and prove an induction-on-scales bound via Fourier analysis. The results illuminate the multi-scale structure of $p$-adic incidence geometry and provide explicit regime-dependent bounds that interpolate between Euclidean and finite-field behavior, with potential applications in discrete geometry over rings and related harmonic analysis on $p$-adic spaces.

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.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof : Proof of Lemma \ref{['sepintersec']}
• proof : Proof of Theorem \ref{['BKTanalogueThm']}
• Lemma 2.2
• proof
• Lemma 2.3
• proof : Proof of Theorem \ref{['SZp-adicThm']}