Spherical Recurrence and locally isometric embeddings of trees into positive density subsets of $\mathbb{Z}^d$

TL;DR

This paper extends results on large density subsets of integer lattices by showing they contain isometric embeddings of trees with prescribed edge lengths, using ergodic theory techniques.

Contribution

It introduces ergodic theoretic methods to prove that dense subsets of  contain locally isometric copies of all trees with edges in a fixed arithmetic progression.

Findings

Dense subsets contain all chains with gaps in a fixed arithmetic progression.

The techniques connect ergodic theory with combinatorial geometry.

Results generalize previous work on distances to more complex tree structures.

Abstract

Magyar has shown that if $B \subset \mathbb{Z}^d$ has positive upper density $(d \geq 5)$, then the set of squared distances $\{ \|b_1-b_2 \|^2 \text{ }: \text{ } b_1,b_2 \in B \}$ contains an infinitely long arithmetic progression, whose period depends only on the upper density of $B$. We extend this result by showing that $B$ contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of $B$ and the number of vertices of the sought tree). In particular, $B$ contains all chains of elements with gaps in some given arithmetic progression (which depends on the length of the sought chain). This is a discrete analogue of a result obtained recently by Bennet, Iosevich and Taylor on chains with prescribed gaps in sets of large Haussdorf dimension. Our techniques are Ergodic theoretic and may be of independent interest to Ergodic theorists. In particular, we obtain Ergodic theoretic analogues of recent \textit{optimal spherical distribution} results of Lyall and Magyar which, via Furstenberg's correspondence principle, recover their combinatorial results.