  We study word metrics on Z^d by developing tools that are fine enough to
measure dependence on the generating set. We obtain counting and distribution
results for the words of length n. With this, we show that counting measure on
spheres always converges to a limit measure on a limit shape (strongly, in an
appropriate sense). The existence of a limit measure is quite strong-even
virtually abelian groups need not satisfy these kinds of asymptotic formulas.
Using the limit measure, we can reduce probabilistic questions about word
metrics to problems in convex geometry of Euclidean space. As an application,
we give asymptotics for the spherical growth function with respect to any
generating set, as well as statistics for other "size-like" functions.
