Improved Bounds on Sidon Sets via Lattice Packings of Simplices

A Bh set (or Sidon set of order h) in an Abelian group G is any subset {b0, b1, . . . , bn} ⊂ G with the property that all the sums bi1 + · · ·+ bih are different up to the order of the summands. Let φ(h, n) denote the order of the smallest Abelian group containing a Bh set of cardinality n+1. It is shown that, as h → ∞ and n is kept fixed, φ(h, n) ∼ 1 n! δl(△) h n , where δl(△ ) is the lattice-packing density of an n-simplex in the Euclidean space. This determines the asymptotics exactly in cases where this density is known (n ≤ 3), and gives an improved upper bound on φ(h, n) in the remaining cases. Covering analogs of Sidon sets are also introduced and their characterization in terms of lattice-coverings by simplices is given. 1. Preliminaries 1.1. Bh sets. Let G be a finite additive Abelian group, and h and n positive integers. A set B = {b0, b1, . . . , bn} ⊂ G is said to be a Bh set (or Sidon set of order h) if all the sums bi1 + · · · + bih with 0 ≤ i1 ≤ · · · ≤ ih ≤ n are different. If B is a Bh set, then so is B − b0 ≡ {0, b1 − b0, . . . , bn − b0}, and vice versa; therefore, we shall assume in the sequel that b0 = 0. With this convention B is a Bh set if and only if the sums α1b1 + · · ·+ αnbn are different for every choice of the coefficients αi ∈ Z, αi ≥ 0, ∑ n i=1 αi ≤ h (here αibi denotes the sum of αi copies of the element bi ∈ G). Let φ(h, n) denote the order of the smallest Abelian group containing a Bh set of cardinality n + 1. Understanding the dependence of φ(h, n) on the parameters h and n, and in particular its asymptotic behavior in various regimes when h·n → ∞, are among the most important problems in the study of Bh sets. We are interested here in the behavior of φ(h, n) as h → ∞ and n is kept fixed. The following bounds are known: (1.1) (2n)! 2n(n!)3 (h− 2n+ 2) < φ(h, n) ≤ (h+ 1), where the left-hand inequality [9] holds for 0 ≤ 2n− 2 ≤ h, and the right-hand inequality [8] for all positive h, n. In particular, (1.2) (2n)! 2n(n!)3 ≤ lim h→∞ φ(h, n) h ≤ 1. Date: October 6, 2016. 2010 Mathematics Subject Classification. 05B10, 05B40, 11B75, 11B83, 11H31, 52C17, 20K99.