Multidimensional van der Corput sets and small fractional parts of polynomials

TL;DR

This paper investigates the smallness of fractional parts of generalized polynomials, including non-integer powers and prime sequences, establishing Diophantine inequalities and connecting to classical and recent number theory results.

Contribution

It introduces new Diophantine inequalities for fractional parts of generalized polynomials, extending classical results to non-integer powers and prime sequences.

Findings

Established bounds for fractional parts of sequences like $ u(n)=loor{n^c}+n^k$

Extended Diophantine inequalities to prime sequences

Connected results to classical work of Heilbronn and recent Bergelson et al.

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.}