Problem Statement
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?
Categories:
Geometry Distances
Progress
A problem of Erdős and Sárközi. Erdős [Er77c] writes that 'Sárközi has the sharpest results, but nothing has been published yet'.The trivial upper bound is $O(r)$. Kovac has observed that Sárközy's lower bound in [466] can be adapted to give a lower bound of $\gg r^{0.26}$ for this problem.
See also [465] for a similar problem (concerning upper bounds) and [466] for a similar problem (concerning lower bounds).
Source: erdosproblems.com/953 | Last verified: January 19, 2026