0000000000620103
AUTHOR
Vitaly Bergelson
showing 3 related works from this author
An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications
2020
Author's accepted manuscript. This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Bergelson, V., Knutson, I. J. H. & Son, Y. (2020). An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications. International Mathematics Research Notices, 2021(19), 14965-15018 is available online at: https://academic.oup.com/imrn/article/2021/19/14965/5775499 and https://doi.org/10.1093/imrn/rnaa035. Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking th…
Weak mixing implies weak mixing of higher orders along tempered functions
2009
AbstractWe extend the weakly mixing PET (polynomial ergodic theorem) obtained in Bergelson [Weakly mixing PET. Ergod. Th. & Dynam. Sys.7 (1987), 337–349] to much wider families of functions. Besides throwing new light on the question of ‘how much higher-degree mixing is hidden in weak mixing’, the obtained results also show the way to possible new extensions of the polynomial Szemerédi theorem obtained in Bergelson and Leibman [Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J. Amer. Math. Soc.9 (1996), 725–753].
An extension of Weyl's equidistribution theorem to generalized polynomials and applications
2019
Generalized polynomials are mappings obtained from the conventional polynomials by the use of operations of addition, multiplication and taking the integer part. Extending the classical theorem of H. Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) \lambda)_{n \in \mathbb{Z}}$ is well distributed $\bmod \, 1$ for all but countably many $\lambda \in \mathbb{R}$ if and only if $\lim\limits_{\substack{|n| \rightarrow \infty n \notin J}} |q(n)| = \infty$ for some (possibly empty) set $J$ having zero density in $\mathbb{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension …