On Few-Distance Sets in the Plane

Let $g(k)$ be the maximum size of a planar set that determines at most $k$ distances. We prove $$\frac{\pi }{3C({\mathrm{\Lambda }}_{hex})}k\sqrt{\log k}(1+o(1))\le g(k)\le Ck\log k,$$ so $g(k)\asymp k\sqrt{\log k}$ with an explicit constant from the hexagonal lattice. For any arithmetic lattice $\mathrm{\Lambda }$ we show $$g_{\mathrm{\Lambda }}(k)\ge (\pi \mathrm{/}4)S^{\ast }(\mathrm{\Lambda })k\sqrt{\log k}(1+o(1))\mathrm{.}$$ 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 $2\mathrm{\Lambda }$.