Additive bases in groups

TL;DR

This paper investigates the properties of additive bases in general abelian groups, introducing analogues of classical functions and establishing bounds that depend on the group's structure.

Contribution

It extends classical additive basis concepts to general abelian groups by defining new functions and deriving bounds, highlighting different behaviors across group types.

Findings

Bounds on functions $S_G$ and $E_G$ valid for all abelian groups

Behavior of $X_G$ varies depending on the group structure

New analogues of classical functions introduced for abelian groups

Abstract

In this paper, we study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions $X$, $S$ and $E$ (defined in the case of the integers) and obtain bounds on them. Our estimates on the functions $S_G$ and $E_G$ are valid for general abelian groups $G$ while in the case of $X_G$ we show that distinct types of behaviours may occur depending on the group $G$.