Improved Bounds on Sidon Sets via Lattice Packings of Simplices

TL;DR

This paper establishes precise asymptotic bounds for Sidon sets in Abelian groups by linking their properties to lattice packings of simplices, improving known bounds and providing geometric characterizations.

Contribution

It introduces a novel connection between Sidon sets and lattice packings of simplices, leading to exact asymptotic formulas and improved bounds for the minimal group order containing such sets.

Findings

Exact asymptotic formula for (h,n) as h

Improved bounds on (h,n) for cases where lattice packing density is unknown

Geometric characterization of bases of order h via lattice coverings

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 \[ \lim_{h \to \infty} \frac{ \phi(h,n) }{ h^n } = \frac{1}{n! \delta_L(\triangle^n)} , \] 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.