Year: 2017
Type: article
Abstract: 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 ${\scriptstyle\lim_{h \to \infty} \frac{ \phi(h,n) }{ h^n } = \frac{1}{n! \ \delta_{\sc l}(\triangle^n)}},$ where $ \delta_{\sc 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.
DOI: 10.1137/16m1099182
Authors: Mladen Kovačević, Vincent Y. F. Tan
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G3077877989
OpenAlex ID: W2575096499