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arXiv · 2010.02325

Iterated differences sets, diophantine approximations and applications

Vitaly Bergelson·Rigoberto Zelada
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Abstract

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal R(v,\epsilon)=\{n\in\mathbb{N}\,|\,\|v(n)\|{<\epsilon\}}$ where $\|\cdot\|$ 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-S\'ark\"ozy theorem.

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Vitaly Bergelson, Rigoberto Zelada. 2020-10-05. Iterated differences sets, diophantine approximations and applications. https://doi.org/10.1016/j.jcta.2021.105520

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