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The number theoretic analogue of a net in metric geometry suggests newproblems 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 ofintegers with respect to the generating set A_g = {g^i:i=0,1,2,...} requires aknowledge of the word lengths of integers with respect to A_g. A g-adicrepresentation of an integer is described that algorithmically produces arepresentation of shortest length. Additive complements and additive asymptoticcomplements are also discussed, together with their associated minimalityproblems.

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