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
\[\lim_{x\to \infty}N(X,\delta)=\infty?\]
