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 $$\fracπ{3\,C(Λ_{hex})}\ k\sqrt{\log k} (1+o(1)) \le g(k) \le C k\log k,$$ 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