Spherical Designs and Generalized Sum-Free Sets in Abelian Groups

TL;DR

This paper introduces the concept of t-free sets in abelian groups, providing bounds for their sizes and demonstrating how they can be used to construct spherical t-designs, extending classical combinatorial number theory concepts.

Contribution

It defines t-free sets in abelian groups, derives asymptotic bounds for their sizes, and links these sets to the construction of spherical t-designs.

Findings

Asymptotic bounds for largest t-free sets in Z_n

Construction methods for spherical t-designs from t-free sets for t ≤ 3

Extension of sum-free set concepts to spherical design applications

Abstract

We extend the concepts of sum-free sets and Sidon-sets of combinatorial number theory with the aim to provide explicit constructions for spherical designs. We call a subset $S$ of the (additive) abelian group $G$ {\it $t$-free} if for all non-negative integers $k$ and $l$ with $k+l \leq t$, the sum of $k$ (not necessarily distinct) elements of $S$ does not equal the sum of $l$ (not necessarily distinct) elements of $S$ unless $k=l$ and the two sums contain the same terms. Here we shall give asymptotic bounds for the size of a largest $t$-free set in ${\bf Z}_n$, and for $t \leq 3$ discuss how $t$-free sets in ${\bf Z}_n$ can be used to construct spherical $t$-designs.