Equidistribution in the $d$-dimensional $a$-adic solenoids

Given a probability measure $mu$ on the $d$-dimensional $a$-adic solenoid $\Omega_a^d$ and an endomorphism $T$ of $\Omega_a^d,$ we consider the relation between uniform distribution of the sequence $T^n\mathbf{x}$ for $\mu$-almost all $\mathbf{x}\in\Omega_a^d$ and the behavior of $\mu$ relative to the translations by some rational subgroups of $\Omega_a^d.$ The main result of this note is an extension of the corresponding result for the $d$-dimensional torus $\mathbb T^d$ due to B. Host \cite{H}.