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 of classical results of I. Vinogradov and G. Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.

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Weyl 等分分布定理在广义多项式中的推广及应用

广义多项式是通过使用加法、乘法和取整数部分的操作从常规多项式获得的映射。扩展 H. Weyl 关于多项式等分布的经典定理，我们证明广义多项式 $q(n)$ 具有序列 $(q(n) \lambda)_{n \in \mathbb{Z} }$ 分布良好 $\bmod \, 1$ 对于几乎所有的 $\lambda \in \mathbb{R}$ 当且仅当 $\lim\limits_{\substack{|n| \rightarrow \infty n \notin J}} |q(n)| = \infty$ 对于某些（可能是空的）集合 $J$ 在 $\mathbb{Z}$ 中具有零密度。我们还沿着素数证明了这个定理的一个版本（这可以看作是 I. Vinogradov 和 G. Rhin 的经典结果的扩展）。最后，我们利用这些结果来获得递归集和范德语料库集的新示例。