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

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.

💡 Research Summary

The paper introduces a number‑theoretic analogue of the geometric notion of a net and applies it to the additive group of integers ℤ equipped with the generating set
(A_g={g^i\mid i=0,1,2,\dots}) for a fixed integer (g>1). An “(h)-net” in this context is a subset (N\subseteq\mathbb Z) such that every integer can be expressed as a sum of at most (h) elements of (A_g) (allowing signs). The central problem is to determine, for any integer (n), the minimal number of generators from (A_g) needed to represent (n); this number is the word length of (n) with respect to (A_g).

To answer this, the authors develop a “(g)-adic representation” that yields the shortest possible expression. Classical base‑(g) expansions use digits in (