A simultaneous version of Host's equidistribution Theorem

Abstract

Let $\mu$ be a probability measure on $\mathbb{R}/\mathbb{Z}$ that is ergodic under the $\times p$ map, with positive entropy. In 1995, Host showed that if $\gcd (m,p)=1$ then $\mu$ almost every point is normal in base $m$. In 2001, Lindenstrauss showed that the conclusion holds under the weaker assumption that $p$ does not divide any power of $m$. In 2015, Hochman and Shmerkin showed that this holds in the "correct" generality, i.e. if $m$ and $p$ are independent. We prove a simultaneous version of this result: for $\mu$ typical $x$, if $m>p$ are independent, we show that the orbit of $(x,x)$ under $(\times m,\times p)$ equidistributes for the product of the Lebesgue measure with $\mu$. We also show that if $m>n>1$ and $n$ is independent of $p$ as well, then the orbit of $(x,x)$ under $(\times m,\times n)$ equidistributes for the Lebesgue measure.