# A note on sets avoiding rational distances

## TL;DR

This paper provides a concise proof that within any subset of real numbers, there exists a large subset avoiding rational distances, and extends some results to subsets of the plane under certain conditions.

## Contribution

It offers a simplified proof of a known result and constructs a Bernstein subset of the reals that avoids rational distances, extending to measurable plane subsets.

## Key findings

- Existence of full subsets avoiding rational distances in any real subset
- Construction of a Bernstein subset that avoids rational distances
- Extension of results to measurable subsets of the plane

## Abstract

In this paper we shall give a short proof of the result originally obtained by Ashutosh Kumar that for each $A\subset \mathbb{R}$ there exists $B\subset A$ full in $A$ such that no distance between two distinct points from $B$ is rational. We will construct a Bernstein subset of $\mathbb{R}$ which also avoids rational distances. We will show some cases in which the former result may be extended to subsets of $\mathbb{R}^2$, i. e. it remains true for measurable subsets of the plane and if $non(\mathcal{N})=cof(\mathcal{N})$ then for a given set of positive outer measure we may find its full subset which is a partial bijection and avoids rational distances.