Asymptotic optimality of the integer-lattice construction

Establish whether the integer-lattice construction is asymptotically optimal for the Erdős distinct distances problem in every dimension, meaning that an N-point configuration in R^d must determine, up to asymptotic order, at least as many distinct distances as the corresponding section of the integer lattice.

Background

The paper introduces the Erdős distinct distances problem as the problem of minimizing the number of pairwise Euclidean distances determined by N points in R^d. A section of the integer lattice provides configurations with approximately N/\sqrt{\log N} distances in dimension two and approximately N^{2/d} distances in dimensions d\geq 3.

The authors note that the lattice construction is known to be near-optimal in several settings, including the plane up to logarithmic factors and R^3 up to a subpolynomial factor. However, the full asymptotic optimality assertion remains broader than the R^3 result proved in the paper.