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 …