How to prove that the distance $|x-y|$ in $A$ (where $A$ is a subset of $\mathbb{R}$ with positive lebesgue measure) does not belong in the Cantor set

Question

Obviously if their distance is above 1 they don’t, but I’m curious on how to approach this for $|x-y|<1$. A thought is to prove that for this set $A$ there exist two elements $x$ and $y$ whose distance is a number that has 1 in its base 3 expansion, but I don’t know how to do that. Is there perhaps another way? I’m thinking proof by contradiction but I’m not sure…