SOLVED

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 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 $\delta>0$) \[N(X,\delta)<X^{1/2+o(1)}?\]

#465: [ErGr80][Er82e]