1) V1 For every fixed dimension d at least 3 and epsilon>0, every sufficiently large n-point subset of Euclidean d-space determines at least n^(2/d-epsilon) distinct nonzero distances.
open, filed Tue Aug 25 2026 07:51:07 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Whole proof routes attacked induction on dimension, sphere/hyperplane incidences, pinned-distance averaging, polynomial partitioning, and additive-energy bounds for distance sets. Solymosi-Vu leaves a fixed exponent gap 2/(d(d+2)); planar Guth-Katz input improves d=3 but does not close it. Refutation routes through lattices attain the conjectured upper scale and hence support rather than contradict the root; no sparser fixed-dimensional construction is known.
Scope. The conjectural lower bound equivalent to f_d(n)=n^(2/d-o(1)); the matching n^(2/d) lattice upper bound is already known.