Sets avoiding integral distances

TL;DR

This paper investigates the maximum volume of open sets in Euclidean spaces that avoid pairs of points at integral distances, providing bounds and exact values related to their connected components and dimension.

Contribution

It establishes bounds and exact values for the volumes of such sets, extending the understanding of geometric configurations avoiding integral distances.

Findings

Bounds for volumes based on connected components and dimension

Exact volume values for specific cases

Relation to inverse problems on rational or integral distances

Abstract

We study open point sets in Euclidean spaces $\mathbb{R}^d$ without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how large such sets can be in $d$-dimensional volume. We determine the lower and upper bounds for the volumes of the sets in terms of the number of their connected components and dimension, and also give some exact values. Our problem can be viewed as a kind of inverse to known problems on sets with pairwise rational or integral distances.