Host equidistribution theorem

Let $p,q\ge 2$ be relatively prime integers. If $\nu$ is invariant and ergodic for the integer multiplication map on the circle $T_p$ and $h_{\nu }(T_p)>0$, then for $\nu$-almost every $x$ the sequence $(T_q^{n}x)$ is an equidistributed sequence for Lebesgue measure. For a non-ergodic $T_p$ invariant measure, the same conclusion holds if almost every measure in its ergodic decomposition has positive entropy. Positive entropy of the whole measure alone does not suffice.

A probability measure on the circle invariant under $T_2$ and $T_3$ and ergodic for their jointly generated semigroup is Lebesgue measure if either transformation has positive Kolmogorov-Sinai entropy. Without joint ergodicity, a mixture of Lebesgue measure and an atomic invariant measure shows why positive entropy alone is insufficient.