Spherical Designs and Generalized Sum-Free Sets in AbelianGroups

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 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 \small{{ Z}}_n, and for t \leq 3 discuss how t-free sets in \small{{ Z}}_n can be used to construct spherical t-designs.