Equidistribution in the dual group of the S-adic integers

Let X be the quotient group of the S-adele ring of an algebraic number field by the discrete group of S-integers. Given a probability measure µ on Xd and an endomorphism T of Xd, we consider the relation between uniform distribution of the sequence Tnx for µ-almost all x ∈ Xd and the behavior of µ relative to the translations by some rational subgroups of Xd. The main result of this note is an extension of the corresponding result for the d-dimensional torus Td due to B.Host.