  We use Nathanson's $g$-adic representation of integers to relate metric
properties of Cayley graphs of the integers with respect to various infinite
generating sets $S$ to problems in additive number theory. If $S$ consists of
all powers of a fixed integer $g$, we find explicit formulas for the smallest
positive integer of a given length. This is related to finding the smallest
positive integer expressible as a fixed number of sums and differences of
powers of $g$. We also consider $S$ to be the set of all powers of all primes
and bound the diameter of Cayley graph by relating it to Goldbach's conjecture.
