Problem 953

Erdős Problem #953: declared status 'open'. Formalized: no. Let $A\subset \{ x\in \mathbb{R}^2 : \lvert x\rvert <r\}$ be a measurable set with no integer distances, that is, such that $\lvert a-b\rvert \not\in \mathbb{Z}$ for any distinct $a,b\in A$. How large can the measure of $A$ be? Current best: The trivial upper bound is $O(r)$. Kovac has observed that S\'{a}rk\"{o}zy's lower bound in [466] can be adapted to give a lower bound of $\gg r^{0.26}$ for this problem. Prize: no. Tags: distances, geometry.