Erdős Problem #466

Let \(N(X,\delta)\) denote the maximum number of points \(P_1,\ldots,P_n\) which can be chosen in a circle of radius \(X\) such that \[\| \lvert P_i-P_j\rvert \| \geq \delta\] for all \(1\leq i<j\leq n\). (Here \(\|x\|\) is the distance from \(x\) to the nearest integer.) Is there some \(\delta>0\) such that \[\lim_{X\to \infty}N(X,\delta)=\infty?\] Graham proved this is true, and in fact \(N(X,1/10)> \frac{\log X}{10}\). This was substantially improved by Sárközy [Sa76], who proved that for all sufficiently small \(\delta>0\), \(N(X,\delta)>X^{1/2-\delta^{1/7}}\).