  We investigate the dimension of the set of points in the d-torus which have
the property that their orbit under rotation by some alpha hits a fixed closed
target A more often than expected for all finite initial portions. An upper
bound for the lower Minkowski dimension of this "strictly heavy set" H(A,alpha)
is found in terms of the upper Minkowski dimension of the boundary of A, as
well as k, the Diophantine approximability from below of the Lebesgue measure
of A. The proof extends to translations in compact abelian groups more
generally than just the torus, most notably the p-adic integers.
