On Few-Distance Sets in the Plane

2025/10/10 by Wang, Lucas
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2510.09800

Abstract

Let g(k) be the maximum size of a planar set that determines at most k distances. We prove \fracπ3 C(Λ_hex) k√(log k) (1+o(1)) ≤ g(k) ≤ C klog k, so g(k) \asymp k√(log k) with an explicit constant from the hexagonal lattice. For any arithmetic lattice Λ we show g_Λ(k)≥ (π/4) S^*(Λ) k√(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∩ W lies in one residue class modulo 2Λ.

Citations

• Incidences in Three Dimensions and Distinct Distances in the Plane

Related

• Lattice Configurations Determining Few Distances
• The cylindrical width of transitive sets
• Semidefinite programming bounds for few-distance sets in the Hamming and Johnson spaces
• Constructions of maximum few-distance sets in Euclidean spaces
• The density of planar sets avoiding unit distances