Multidimensional van der Corput sets and small fractional parts of polynomials

Abstract

We establish Diophantine inequalities for the fractional parts of generalized polynomials $f$, in particular for sequences $\nu (n)=\lfloor n^c\rfloor +n^k$ with $c>1$ a non-integral real number and $k\in \mathbb{N}$, as well as for $\nu (p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson \textit{et al.}

26 pages