Necessity of lattice structure in asymptotically optimal configurations

Determine whether lattice structure is necessary, in an asymptotic sense, for point configurations that attain the minimum possible number of distinct distances in the Erdős distinct distances problem.

Background

The introduction discusses a broader belief in discrete geometry that extremal configurations often have the global structure of a lattice, with the integer lattice serving as an asymptotically optimal example. It also notes that alternative lattices can perform comparably or, in some related problems, outperform the integer lattice.

The unresolved issue is not whether lattice examples exist, but whether every asymptotically optimal configuration must exhibit lattice structure. The paper explicitly records that this necessity has not been established.