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

Abstract

Let $A=\{a_1,a_2,\dots ,a_m\}$ be a subset of a finite abelian group $G$. We call $A$ {\it $t$-independent} in $G$, if whenever $${\lambda }_1{a}_1+{\lambda }_2{a}_2+\cdots +{\lambda }_m{a}_m=0$$ for some integers ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$ with $$|{\lambda }_1|+|{\lambda }_2|+\cdots +|{\lambda }_m|\le t,$$ we have ${\lambda }_1={\lambda }_2=\cdots ={\lambda }_m=0$, and we say that $A$ is {\it $s$-spanning} in $G$, if every element $g$ of $G$ can be written as $$g={\lambda }_1{a}_1+{\lambda }_2{a}_2+\cdots +{\lambda }_m{a}_m$$ for some integers ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$ with $$|{\lambda }_1|+|{\lambda }_2|+\cdots +|{\lambda }_m|\le s.$$ 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.