Bi-invariant sets and measures have integer Hausdorff dimension

Let A, B be two diagonal endomorphisms of the d-dimensional torus with corr esponding eigenvalues relatively prime. We show that for any A-invariant er godic measure mu, there exists a projection onto a torus T-r of dimension r greater than or equal to dim mu, that maps mu-almost every B-orbit to a un iformly distributed sequence in T-r. As a corollary we obtain that the Haus dorff dimension of any bi-invariant measure, as well as any closed bi-invar iant set, is an integer.