OPEN
This is open, and cannot be resolved with a finite computation.
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?
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).
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This page was last edited 16 September 2025.
Additional thanks to: Vjekoslav Kovac
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #953, https://www.erdosproblems.com/953, accessed 2026-01-16
