Maximum measure of planar sets avoiding integer distances
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Let M(R) be the largest measure of a measurable subset of a radius-R planar disk containing no positive integer distance between distinct points. What is its growth exponent as R tends to infinity?
Why it matters and what counts as progress
Why it matters: Matching exponents would determine the measure growth of large planar sets avoiding every positive integer distance.
What counts as progress: Match the known lower exponent with an O(√R) upper bound or sharpen subpower factors in the measurable problem.
How the problem developed
- 1977 · Problem [1]
Paul Erdős and András Sárközy Posed the extremal problem of bounding the measure of planar sets that avoid integer distances.
- 27 Apr 2026 · Proof [1]
Przemek Chojecki, with GPT-5.5 Pro Announced the upper bound M(R) ≪ R¹ᐟ²; combined with Sárközy’s lower bound, this gives M(R)=R¹ᐟ²⁺ᵒ⁽¹⁾.
- 1 May 2026 · Human check [1]
Nat Sothanaphan Reported that a standard check of the short proof found no issue.
AI contribution
GPT-assisted bound settles Erdős #953 exponent
For the largest measure M(R) of a measurable subset of a radius-R planar disk with no positive integer distance between distinct points, M(R)≤C√R. Together with Sárközy’s earlier lower construction, this yields M(R)=R^(1/2+o(1)).
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