  In a previous paper [arXiv:1006.3807], the authors obtained tube formulas for
certain fractals under rather general conditions. Based on these formulas, we
give here a characterization of Minkowski measurability of a certain class of
self-similar tilings and self-similar sets. Under appropriate hypotheses,
self-similar tilings with simple generators (more precisely, monophase
generators) are shown to be Minkowski measurable if and only if the associated
scaling zeta function is of nonlattice type. Under a natural geometric
condition on the tiling, the result is transferred to the associated
self-similar set (i.e., the fractal itself). Also, the latter is shown to be
Minkowski measurable if and only if the associated scaling zeta function is of
nonlattice type.
