= Host equidistribution theorem
{c}
Let $p,q\geq2$ 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 $(T_q^nx)$ 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.
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