Heaviness in Toral Rotations

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.