On a theorem of Mattila in the p-adic setting

TL;DR

This work analyzes a p-adic analogue of Mattila’s incidence problem in the plane over ${\mathbb Z}/p^r{\mathbb Z}$. Employing discrete Fourier analysis and restriction/extension theory for p-adic circles, it proves a sharp density threshold: for ${p\equiv 3\pmod{4}}$, if ${\delta_A^{1/2}\delta_B \ge 2 p^{-1}}$, then a positive proportion of rotations ${g\in SO_2({\mathbb Z}/p^r{\mathbb Z})}$ satisfy ${|gA-B| \gg p^{2r}}$; a product-density condition suffices when ${p\equiv 1\pmod{4}}$. The paper establishes universal and refined incidence bounds between points and rigid motions, uses them to bound the key energy sums, and discusses sharpness via coset constructions and open questions for balanced densities and higher dimensions. This advances the understanding of rigid-motion incidence phenomena in non-Euclidean finite rings and connects to Mattila’s classical results and finite-field analogues.

Abstract

Let $A, B$ be subsets of $(\mathbb{Z}/p^r\mathbb{Z})^2$. In this note, we provide conditions on the densities of $A$ and $B$ such that $|gA-B|\gg p^{2r}$ for a positive proportion of $g\in SO_2(\mathbb{Z}/p^r\mathbb{Z})$. The conditions are sharp up to constant factors in the unbalanced case, and the proof makes use of tools from discrete Fourier analysis and results in restriction/extension theory.

Theorems & Definitions (31)

• Example 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Lemma 2.1: P.X, Lemma 4.5
• Lemma 2.2: P.X, Lemma 4.9
• Theorem 2.3: P.X, Theorem 4.1
• Theorem 2.4: P.X, Theorem 4.2
• Remark 2.5
• Lemma 2.6