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

Abstract

Let A = {a ₁, a ₂, …, a _m } be a subset of a finite abelian group G. We call A t-independent in G, if whenever

$$ \lambda _1 a_1 + \lambda _2 a_2 + ... + \lambda _m a_m = 0 $$

for some integers λ₁, λ ₂,…, λ _m with

$$ |\lambda _1 | + \lambda _2 | + ... + |\lambda _m | \leqslant t, $$

we have λ ₁ = λ ₂ = … = λ _m = 0, and we say that A is s-spanning in G, if every element g of G can be written as

$$ g = \lambda _1 a_1 + \lambda _2 a_2 + ...\lambda _m a_m $$

for some integers λ ₁, λ ₂, …, λ _m with

$$ |\lambda _1 | + |\lambda _2 | + ... + |\lambda _m | \leqslant s. $$

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.