Erdős problem 953

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