On Few-Distance Sets in the Plane
Lucas Wang
TL;DR
This work determines the growth of g(k), the maximal size of a planar point set determining at most k distances, up to explicit constants. By blending incidence geometry (Guth–Katz), additive combinatorics (BSG–Freiman toolkit), and lattice-geometry via lattice windows, it proves g(k) ≍ k √log k with an explicit hexagonal-lattice constant for the lower bound and a universal upper bound of O(k log k). For arithmetic lattices Λ, it establishes a matching lower bound g_Λ(k) ≳ (π/4) S^*(Λ) k √log k (1+o(1)). A quantitative stability theorem is also proved: unless the configuration is line-heavy or has two popular nonparallel shifts, extremizers exhibit either near-center localization or concentration in a single residue class modulo 2Λ, revealing a rigid geometric structure for near-optimizers.
Abstract
Let $g(k)$ be the maximum size of a planar set that determines at most $k$ distances. We prove $$\fracπ{3\,C(Λ_{hex})}\ k\sqrt{\log k} (1+o(1)) \le g(k) \le C k\log k,$$ so $g(k) \asymp k\sqrt{\log k}$ with an explicit constant from the hexagonal lattice. For any arithmetic lattice $Λ$ we show $$g_Λ(k)\ge (π/4) S^*(Λ) k\sqrt{\log k} (1+o(1)).$$ 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 $2Λ$.