Moree 2006 two dimensional lattices few distances

../

theorem_1: Every planar lattice not isometric to the normalized hexagonal lattice has Erdős number above 0.5533117758..., and below any bound r only finitely many lattices remain up to homothety, all explicitly determinable.

theorem_5: Gives the number g(n, D) of genera of discriminant D that represent n: zero unless (n, f^2) is a square and no prime p with Kronecker symbol (d_0/p) = -1 divides n to an odd power, and otherwise 2^{t(D) - t(D/(n, f^2))}.

---

Moree, Pieter and Osburn, Robert, Two-dimensional lattices with few distances. Enseign. Math. (2) 52 (2006), 361--380.

Theorem 1 (p. 2 of the arXiv version) shows that any two-dimensional lattice L not isometric to the normalized hexagonal lattice Sigma has Erdos number E_L strictly greater than E_Sigma = 2^{-3/2} 3^{1/4} prod_{p = 2 mod 3} (1-1/p^2)^{-1/2} = 0.5533117758..., so the hexagonal lattice asymptotically determines the fewest distances; the theorem also shows that for any real r the set of non-homothetic lattices with E_L < r is finite and explicitly determinable. Here the Erdos number is E_L = F_L d^{1/2}, where d is the determinant of L and the population fraction F_L is the limit of N_L(x) sqrt(log x)/x, N_L(x) counting the distinct values up to x that the associated binary quadratic form takes. This closes the n = 2 case, which Conway and Sloane's 1991 work, settling dimensions 3 to 8, claimed on the strength of a never-published 1990 preprint of W. D. Smith. The proof combines an explicit formula for the number of genera of discriminant D representing an integer (Theorem 5) with a recent improved lower bound for Euler's phi function at odd arguments, and the paper surveys related literature including Schmutz Schaller's stronger 1995 conjecture (Conjecture 1) that in dimensions 2 to 8 the even lattices of minimal determinant have maximal length spectra. For problem 659 this supplies lattice-distance-counting background. The paper's covolume-one optimization does not impose the additional condition that every four points determine at least three distances, so it does not by itself solve that problem.

Source: https://arxiv.org/abs/math/0604163. The copy read for this card is arXiv version 2 (20 October 2006), which the arXiv record's comment calls the final version, accepted for publication in Enseign. Math. The arXiv record carries no license field, so arXiv's assumed license applies (arXiv:math/0604163), every other right reserved.

Bears on. #659: background only. For a lattice with finite Erdos number the count of distinct squared distances up to x grows like a constant times x/sqrt(log x), and Theorem 1 names the lattice with the least normalized constant. The paper imposes no four-point condition, and its minimizer, the hexagonal lattice, has four points (two equilateral triangles sharing an edge) with only two distances.

Results.

• Theorem 1 (p. 2): every planar lattice not isometric to Sigma has E_L > E_Sigma = 0.553311775832479...; for each real r the lattices with E_L < r are finitely many up to homothety and explicitly determinable.
• Theorem 5 (p. 11): the number g(n, D) of genera of discriminant D representing n, which the paper obtains from results of Kaplan and Williams and of Sun and Williams and uses as an input to Theorem 1.

Conjecture 1 (p. 3) is Schmutz Schaller's, surveyed rather than proved here: in dimensions 2 to 8 the even lattices with minimal determinant have "maximal lengths", their length spectrum dominating that of every other lattice of the same dimension and covolume at every position. Page numbers on this card and its result pages are those of the arXiv version read.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.