Iterated differences sets, diophantine approximations and applications

Abstract

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 $\mathcal{R}(v,\epsilon )=\{n\in \mathbb{N}|\Vert v(n)\Vert <\epsilon \}$ 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.

43 pages, referees' comments included, minor errors in the proofs of Theorem 4.1 and Lemma 5.3 were corrected