A finite subset X on the unit sphere Sd in Rd+1 is a spherical t-design if for every polynomial f: Sd -%gt; R of degree at most t, the average value of f over Sd equals the average of f on X. Spherical designs have been studied extensively via combinatorics, approximation theory, and other fields.
Let G be an additive abelian group. We say that S (a subset of G) is a t-independent set in G if for all non-negative integers k and l with k+l <= 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 the two sums contain the same terms. This concept extends the well studied concepts of sum-free sets and Sidon sets.
In this talk we give some exact values and asymptotic bounds for the maximum size of a t-independent set in the cyclic group and in other abelian groups. As an application, we show how 3-independent sets can be used to construct spherical 3-designs.