PROVED
This has been solved in the affirmative.
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 $\delta>0$)\[N(X,\delta)<X^{1/2+o(1)}?\]
The first conjecture was proved by Sárközy
[Sa76], who in fact proved\[N(X,\delta) \ll \delta^{-3}\frac{X}{\log\log X}.\]Konyagin
[Ko01] proved the strong upper bound\[N(X,\delta) \ll_\delta N^{1/2}.\]See also
[466] for lower bounds and
[953] for a similar problem.
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This page was last edited 16 September 2025.
Additional thanks to: Stefan Steinerberger
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