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

Let $A={a1,a2,\dots, am}$ be a subset of a finite abelian group $G$. We call $A$ {\it $t$-independent} in $G$, if whenever $$\lambda1a1+\lambda2a2+\cdots +\lambdam am=0$$ for some integers $\lambda1, \lambda2, \dots , \lambdam$ with $$|\lambda1|+|\lambda2|+\cdots +|\lambdam| \leq t,$$ we have $\lambda1=\lambda2= \cdots = \lambdam=0$, and we say that $A$ is {\it $s$-spanning} in $G$, if every element $g$ of $G$ can be written as $$g=\lambda1a1+\lambda2a2+\cdots +\lambdam am$$ for some integers $\lambda1, \lambda2, \dots , \lambdam$ with $$|\lambda1|+|\lambda2|+\cdots +|\lambdam| \leq s.$$ 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.