Improved Bounds on Sidon Sets via Lattice Packings of Simplices

A $B_h$ set (or Sidon set of order $h$) in an Abelian group $G$ is any subset $\{b_0, b_1, \ldots,b_{n}\}$ of $G$ with the property that all the sums $b_{i_1} + \cdots + b_{i_h}$ are different up to the order of the summands. Let $\phi(h,n)$ denote the order of the smallest Abelian group containing a $B_h$ set of cardinality $n + 1$. It is shown that $$\lim_{h \to \infty} \frac{ \phi(h,n) }{ h^n } = \frac{1}{n! \delta_L(\triangle^n)} ,$$ where $\delta_L(\triangle^n)$ is the lattice packing density of an $n$-simplex in Euclidean space. This determines the asymptotics exactly in cases where this density is known ($n \leq 3$) and gives improved bounds on $\phi(h,n)$ 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.