Let $N(X,\delta )$ denote the maximum number of points $P_1,\dots ,P_n$ which can be chosen in a circle of radius $X$ such that$$\Vert |P_i-P_j|\Vert \ge \delta$$for all $1\le i<j\le n$. (Here $\Vert x\Vert$ is the distance from $x$ to the nearest integer.)Is there some $\delta >0$ such that$$\underset{x\to \infty }{\lim }N(X,\delta )=\infty ?$$