Heaviness in toral rotations

Abstract

We investigate the dimension of the set of points H(A, α) in the d-torus which have the property that their orbit under rotation by some α hits a fixed closed target A more often than expected for all finite initial portions of time. An upper bound for the lower Minkowski dimension of this set is found in terms of the upper Minkowski dimension of ∂A:

$$\underline {\dim } _M (H(A,\alpha )) \leqslant \frac{{d + \overline {\dim } _M \partial {\rm A}}} {2},$$

which may be improved under certain Diophantine conditions on µ(A). In particular, in the event that µ(A) ∈ ℚ we have

$$\underline {\dim } _M (H(A,\alpha )) \leqslant \overline {\dim } _M \partial {\rm A}.$$

The proof extends to translations in compact abelian groups more generally than just the torus, most notably the p-adic integers.