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Erdős problem 953

What is the largest possible measure of a subset of a radius-RR disk in R2\mathbb{R}^2 containing no pair of points at a positive integer distance? A Poisson-Bessel kernel argument gives M(R)R1/2M(R) \ll R^{1/2}; with Sárközy's lower construction, M(R)=R1/2+o(1)M(R) = R^{1/2 + o(1)}.

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