A simultaneous version of Host’s equidistribution Theorem

Let μ be a probability measure on R/Z that is ergodic under the ×p map, with positive entropy. In 1995, Host showed that if gcd(m, p) = 1, then μ 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 μ typical x, if m > p are independent, we show that the orbit of (x, x) under (×m, ×p) equidistributes for the product of the Lebesgue measure with μ. We also show that if m > n > 1 and n is independent of p as well, then the orbit of (x, x) under (×m, ×n) equidistributes for the Lebesgue measure.