Nets in groups, minimum length $g$-adic representations, and minimal additive complements

TL;DR

This paper explores the concept of nets in groups through g-adic representations, analyzing minimal additive complements and their properties, connecting number theory with geometric and combinatorial ideas.

Contribution

It introduces a g-adic representation algorithm for shortest length expressions and investigates minimal additive complements in the context of nets in groups.

Findings

Development of an algorithm for shortest g-adic representations

Analysis of minimal additive complements and their properties

Connections established between number theory and geometric concepts

Abstract

The number theoretic analogue of a net in metric geometry suggests new problems and results in combinatorial and additive number theory. For example, for a fixed integer g > 1, the study of h-nets in the additive group of integers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires a knowledge of the word lengths of integers with respect to A_g. A g-adic representation of an integer is described that algorithmically produces a representation of shortest length. Additive complements and additive asymptotic complements are also discussed, together with their associated minimality problems.