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 it true that, for any $0<\delta <1/2$, we have$$N(X,\delta )=o(X)?$$In fact, is it true that (for any fixed $\delta >0$)$$N(X,\delta )<X^{1/2+o(1)}?$$