Problem Statement
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\[\lim_{x\to \infty}N(X,\delta)=\infty?\]
Is there some $\delta>0$ such that\[\lim_{x\to \infty}N(X,\delta)=\infty?\]
Categories:
Number Theory
Progress
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 $\delta>0$,\[N(X,\delta)>X^{1/2-\delta^{1/7}}.\]See also [465] for upper bounds and [953] for a similar problem.Source: erdosproblems.com/466 | Last verified: January 15, 2026