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A $ B_h $ set (or Sidon set of order $ h $) in an Abelian group $ G $ is anysubset $ \{b_0, b_1, \ldots,b_{n}\} $ of $ G $ with the property that all thesums $ b_{i_1} + \cdots + b_{i_h} $ are different up to the order of thesummands. Let $ \phi(h,n) $ denote the order of the smallest Abelian groupcontaining 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$-simplex in Euclidean space. This determines the asymptotics exactly in caseswhere this density is known ($ n \leq 3 $) and gives improved bounds on $\phi(h,n) $ in the remaining cases. The corresponding geometriccharacterization of bases of order $ h $ in finite Abelian groups in terms oflattice coverings by simplices is also given.

Improved Bounds on Sidon Sets via Lattice Packings of Simplices