Avoiding rational distances

יום א', 16/11/2014 - 10:15

Komjath has asked the following question: Let X be a subset of Euclidean space. Must there exist a subset Y of X such that X and Y have same outer measure and the distance between any two points in Y is irrational? 

We'll show that this is true in dimension one. Our proof relies on some work of Gitik and Shelah on forcings with sigma ideals.