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 ≥ 2, the study of h-nets in the additive group of integers with respect to the generating set Ag = {0}∪{±g : i = 0, 1, 2, . . .} requires a knowledge of the word lengths of integers with respect to Ag. 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. 1. Nets in metric spaces Let (X, d) be a metric space. For z ∈ X and r ≥ 0, the sphere with center z and radius r is the set Sz(r) = {x ∈ X : d(x, z) = r}. The open ball Bz(r) of radius r and center z and the closed ball Bz(r) of radius r and center z are, respectively, Bz(r) = {x ∈ X : d(x, z) ≤ r} = ⋃ r<r Sz(r ) and Bz(r) = {x ∈ X : d(x, z) ≤ r} = ⋃ r≤r Sz(r ). An r-net in (X, d) is a subset C of X such that, for all x ∈ X , there exists z ∈ C with d(x, z) ≤ r. Equivalently, C is an r-net in X if and only if X = ⋃ z∈C Bz(r). Note that X is the unique 0-net in X . The set C is a net in X if C is an r-net for some r ≥ 0. The set C in X is called r-separated if d(z, z) ≥ r for all z, z ∈ C with z 6= z. By Zorn’s lemma, every metric space contains a maximal r-separated set, and a maximal r-separated set is an r-net in X . A minimal r-net in a metric space (X, d) Date: December 2, 2008. 2000 Mathematics Subject Classification. 11A63, 11B13, 11B34, 11B75, 20F65, 51F99, 54E35.