A simultaneous version of Host’s equidistribution Theorem
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- by Amir Algom PDF
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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.References
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Additional Information
- Amir Algom
- Affiliation: Department of Mathematics, the Pennsylvania State University, University Park, Pennsylvania 16802
- MR Author ID: 1292097
- Email: aka5983@psu.edu
- Received by editor(s): April 29, 2019
- Received by editor(s) in revised form: December 6, 2019
- Published electronically: September 29, 2020
- Additional Notes: The author was supported by ERC grant 306494 and ISF grant 1702/17.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 8439-8462
- MSC (2010): Primary 11K16, 11A63, 28A80, 28D05
- DOI: https://doi.org/10.1090/tran/8173
- MathSciNet review: 4177264