Let v be an odd real polynomial (i.e. a polynomial of the form ${\sum }_{j=1}^{\ell }{a}_j{x}^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal{R}(v,ϵ)=\{n\in \mathbb{N}|\Vert v(n)\Vert <ϵ\}$ where $\Vert \cdot \Vert$ denotes the distance to the closest integer. We then apply the new Diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sárközy theorem.

中文翻译：

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迭代差分集、丢番图近似和应用

设v是一个奇数实数多项式（即形式为${\sum }_{j=1}^{\ell }{\mathrm{一种}}_j{X}^{2j-1}$）。我们利用迭代差异集来建立关于表单集的新结果$\mathcal{电阻}(v,\epsilon )=\{n\in \mathbb{N}|\Vert v(n)\Vert <\epsilon \}$ 在哪里 $\Vert \cdot \Vert$表示到最近整数的距离。然后我们应用新的丢番图结果来获得遍历理论和组合学的应用。特别是，我们获得了弱混合系统的新特征以及 Furstenberg-Sárközy 定理的新变体。