PROBLEMS IN ADDITIVE NUMBER THEORY, IV: NETS IN GROUPS AND SHORTEST LENGTH g-ADIC REPRESENTATIONS

Abstract

The number theoretic analog 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} ∪ {± gi : 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.