On distances between points on the plane (Q5951091)

Let \(\|x\|\) be the distance from \(x\) to the nearest integer and \(d(P;Q)\) denote the distance between points \(P\) and \(Q\) on the plane. For \(X\geq 1\) and \(0<\delta<1/2\), let \(N(X,\delta)\) denote the maximum number of points \(P_1,\dots,P_n\) that can be chosen in the disk of radius \(X\) so that \(\|d(P_i,P_j)\|\geq\delta\) for all \(1\leq i<j\leq n\). Then the author proves that there exists a number \(C(\delta)\) such that \(N(X,\delta)< C(\delta) X^{1/2}\). This result answers a question due to Erdős and Graham.