Distances and Trees in Dense Subsets of $\mathbb{Z}^d$

TL;DR

This paper provides a new direct proof for the existence of large distances in dense subsets of integer lattices and extends the results to arbitrary trees, using discrete spherical maximal functions.

Contribution

It introduces a novel direct proof technique for discrete distance sets and generalizes the results to include arbitrary trees, enhancing previous Fourier analytic approaches.

Findings

New direct proof for distance set results in $\\mathbb{Z}^d$

Extension of results to arbitrary trees

Establishment of pinned variants using discrete spherical maximal functions

Abstract

In \cite{FKW} Katznelson and Weiss establish that all sufficiently large distances can always be attained between pairs of points from any given measurable subset of $\mathbb{R}^2$ of positive upper (Banach) density. A second proof of this result, as well as a stronger "pinned variant", was given by Bourgain in \cite{B} using Fourier analytic methods. In \cite{M1} the second author adapted Bourgain's Fourier analytic approach to established a result analogous to that of Katznelson and Weiss for subsets $\mathbb{Z}^d$ provided $d\geq 5$. We present a new direct proof of this discrete distance set result and generalize this to arbitrary trees. Using appropriate discrete spherical maximal function theorems we ultimately establish the natural "pinned variants" of these results.