Iterated differences sets, diophantine approximations and applications

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 $mathcal R(v,epsilon)={ninmathbb{N},|,|v(n)|{<epsilon}}$ 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-Sarkozy theorem.