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

TL;DR

This paper investigates the maximum size of t-independent and s-spanning subsets in finite abelian groups, providing bounds and characterizing cases of extremal sizes, with connections to spherical combinatorics.

Contribution

It introduces bounds for the sizes of t-independent and s-spanning sets in finite abelian groups and explores cases of extremal sizes, linking to spherical combinatorics.

Findings

Upper bounds for t-independent sets' size

Lower bounds for s-spanning sets' size

Characterization of extremal cases

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 $$\lambda_1a_1+\lambda_2a_2+\cdots +\lambda_m a_m=0$$ for some integers $\lambda_1, \lambda_2, \dots , \lambda_m$ with $$|\lambda_1|+|\lambda_2|+\cdots +|\lambda_m| \leq t,$$ 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 $$g=\lambda_1a_1+\lambda_2a_2+\cdots +\lambda_m a_m$$ for some integers $\lambda_1, \lambda_2, \dots , \lambda_m$ with $$|\lambda_1|+|\lambda_2|+\cdots +|\lambda_m| \leq s.$$ 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.