Iterated differences sets, Diophantine approximations and applications

Abstract

Let v be an odd real polynomial (i.e. a polynomial of the form ∑j=1ℓajx2j−1). We utilize sets of iterated differences to establish new results about sets of the form R(v,ϵ)={n∈N|‖v(n)‖<ϵ} where ‖⋅‖ 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.