Avoiding Rational Distances

TL;DR

This paper proves that for any set of real numbers, there exists a subset with the same Lebesgue outer measure where all pairwise distances are irrational, highlighting a surprising property of real sets.

Contribution

It introduces a novel method to construct subsets of real numbers with prescribed measure and irrational distances, expanding understanding of measure and distance properties.

Findings

Existence of subsets with same Lebesgue outer measure and only irrational distances.

Construction method applicable to any set of real numbers.

Highlights new interactions between measure theory and metric properties.

Abstract

We show that for any set of reals X there is a subset Y such X and Y have same Lebesgue outer measure and the distance between any two distinct points in Y is irrational.