A set with positive outer measure

Question

Let $A$ be a subset of $\mathbb R$ such that $m^*(A)>0$. I want to show that there exist $x,y\in A$ such that $x-y\in \mathbb{R\setminus Q}$.

Suppose for all $x,y\in A, x-y\in \mathbb Q$. We define $f:A\times A\to \mathbb Q$ by $f(x,y)=x-y$ for all $x,y\in A$. Can I show from here that $A$ is countable?

Answer 1

Suppose for all $x,y\in A$, $x-y\in \mathbb{Q}$ and suppose $A$ is non-empty(otherwise trivially $m^{*}(A)=0$), pick any $x\in A$, then $A\subset x+\mathbb{Q}$ therefore at most countable.

For higher dimension, see All distances are rational prove the set is countable