Largest measure of a bounded planar set with no two points an integer distance apart (Erdős #953)
Statement
Let $A\subset\{x\in\mathbb{R}^2:\lvert x\rvert<r\}$ be a Lebesgue-measurable set that contains no two points at an integer distance apart, i.e. $\lvert a-b\rvert\notin\mathbb{Z}$ for every pair of distinct $a,b\in A$. How large can the measure of $A$ be, as a function of $r$? Determine the maximal measure, or at least its order of growth in $r$.
Acceptance. FULLY RESOLVES: determine the growth order of the maximal measure in $r$ (matching upper and lower bounds up to a factor $r^{o(1)}$), with a complete proof. ADVANCES, either side of the current $\gg_\epsilon r^{1/2-\epsilon}\lesssim M(r)\lesssim O(r)$ bracket, each with a complete proof or a reproducible construction plus certificate: (a) a construction giving measure $\gg r^{1/2+c}$ for some fixed $c>0$; (b) an upper bound $o(r)$ on the maximal measure. Deliver the proof and/or the explicit set together with its measure and a certificate that it avoids every integer distance.
Background
A problem of Erdős and Sárközi; Erdős [Er77c] wrote that 'Sárközi has the sharpest results, but nothing has been published yet.' The trivial upper bound is that the measure is $O(r)$. Koizumi and Kovac observed (in the site comments) that Sárközy's lower-bound method for the companion problem (erdosproblems.com/466) can be adapted to give measure $\gg_\epsilon r^{1/2-\epsilon}$ for every $\epsilon>0$. So the maximal measure lies between $\gg r^{1/2-\epsilon}$ and $O(r)$, and the true exponent is open per erdosproblems.com/953 (fetched 2026-07-21, status 'open', tagged 'geometry | distances'). Related site problems: an upper-bound companion at erdosproblems.com/465 and a lower-bound companion at erdosproblems.com/466. Attacker's tool: explicit constructions of large-measure integer-distance-avoiding sets (radial/annular or Cantor-type designs, or Fourier-analytic constructions) to raise the lower exponent, together with density or Fourier/packing arguments to push the $O(r)$ upper bound down; numerical optimisation over parameterised radial constructions can certify concrete lower bounds for specific $r$.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #953 (T. F. Bloom) | website |
| REF-02 | Erdős Problem #466 — companion lower-bound problem (Sárközy's method) | website |
| REF-03 | Erdős Problem #465 — companion upper-bound problem | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.