Erdős Problem #659 — Point Sets with Few Distances

Erdős posed the problem in 1997. It asks for planar sets with unusually few distances while retaining a strong local four-point condition.

Known results

• Moree and Osburn established the $O(n/\sqrt{\log n})$ distance bound for the stretched lattice.
• Lund and Sheffer independently found the construction and excluded equilateral triangles and squares.
• Perucca classified the six four-point configurations determining only two distances.

January 2026 affirmative solution

A preprint constructs $P_m=\{(x,\sqrt{2}y):0\le x,y\le m-1\}$, proves the four-point condition using Perucca’s classification, and obtains the distance bound from Bernays’ theorem for $u^2+2v^2$. A second preprint gives an independent lattice construction and corroborates the affirmative answer.

Current status (as of April 2026): The problem is resolved by explicit preprint constructions; the four-point condition and $O(n/\sqrt{\log n})$ distance bound are established.