## Abstract

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.

Original language | English |
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Pages (from-to) | 8439-8462 |

Number of pages | 24 |

Journal | Transactions of the American Mathematical Society |

Volume | 373 |

Issue number | 12 |

DOIs | |

State | Published - Dec 2020 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2020 American Mathematical Society

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics