**Problem of the Week**

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**Zassenhaus Conference**

**Hilton Memorial Lecture**

seminars:comb:abstract.200203baj

A finite subset X on the unit sphere S^{d} in R^{d+1} is a *spherical t-design* if for every polynomial f: S^{d} -%gt; R of degree at most t, the average value of f over S^{}d 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.

seminars/comb/abstract.200203baj.txt · Last modified: 2020/01/29 14:03 (external edit)

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