On Few-Distance Sets in the Plane

Abstract

Let $g(k)$ be the maximum size of a planar set that determines at most $k$ distances. We prove $$\frac{\pi }{3C({\Lambda }_{hex})}k\sqrt{\log k}(1+o(1))\le g(k)\le Ck\log k,$$ so $g(k)\asymp k\sqrt{\log k}$ with an explicit constant from the hexagonal lattice. For any arithmetic lattice $\Lambda$ we show $$g_{\Lambda }(k)\ge (\pi /4)S^{\ast }(\Lambda )k\sqrt{\log k}(1+o(1)).$$ We also give quantitative stability: unless $X$ is line-heavy or has two popular nonparallel shifts, either almost all ordered pairs lie below a high quantile of the distance multiset (near-center localization), or a constant fraction of $X\cap W$ lies in one residue class modulo $2\Lambda$.

Error in the proof, claimed asymptotic fails