  We discuss a number of techniques for determining the Minkowski dimension of
bounded subsets of some Euclidean space of any dimension, including: the
box-counting dimension and equivalent definitions based on various box-counting
functions; the similarity dimension via the Moran equation (at least in the
case of self-similar sets); the order of the (box-)counting function; the
classic result on compact subsets of the real line due to Besicovitch and
Taylor, as adapted to the theory of fractal strings; and the abscissae of
convergence of new classes of zeta functions. Specifically, we define
box-counting zeta functions of infinite bounded subsets of Euclidean space and
discuss results pertaining to distance and tube zeta functions. Appealing to an
analysis of these zeta functions allows for the development of theories of
complex dimensions for bounded sets in Euclidean space, extending techniques
and results regarding (ordinary) fractal strings obtained by the first author
and van Frankenhuijsen.
