An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications

Abstract

Abstract Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial $q(n)$ has the property that the sequence $(q(n) \lambda )_{n \in \mathbf{Z}}$ is well-distributed $\bmod \, 1$ for all but countably many $\lambda \in{\mathbf R}$ if and only if $\lim\nolimits _{\substack{|n| \rightarrow \infty \\ n \notin J}} |q(n)| = \infty $ for some (possibly empty) set $J$ having zero natural density in $\mathbf{Z}$. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.