  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.
