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, ldots,bn of G with the property that all the sums bi1 + cdots + 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 [ limh → ∞ frac φ(h,n) hn = frac1n! δL(trianglen) , ] where δL(trianglen) is the lattice packing density of an n -simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known ( n ≤ 3 ) and gives improved bounds on φ(h,n) in the remaining cases. The corresponding geometric characterization of bases of order h in finite Abelian groups in terms of lattice coverings by simplices is also given.