The spanning number and the independence number of a subset of an abelian group

Let A=₁,a₂,, aₘ\ be a subset of a finite abelian group G. We call A { t-independent} in G, if whenever λ₁a₁+λ₂a₂+⋯ +λₘ aₘ=0 for some integers λ₁, λ₂, , λₘ with |λ₁|+|λ₂|+⋯ +|λₘ| ≤ t, we have λ₁=λ₂= ⋯ = λₘ=0, and we say that A is { s-spanning} in G, if every element g of G can be written as g=λ₁a₁+λ₂a₂+⋯ +λₘ aₘ for some integers λ₁, λ₂, , λₘ with |λ₁|+|λ₂|+⋯ +|λₘ| ≤ s. In this paper we give an upper bound for the size of a t-independent set and a lower bound for the size of an s-spanning set in G, and determine some cases when this extremal size occurs. We also discuss an interesting connection to spherical combinatorics.