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MULTIDIMENSIONAL VAN DER CORPUT SETS AND SMALL FRACTIONAL PARTS OF POLYNOMIALS

Part of: Diophantine approximation, transcendental number theory Exponential sums and character sums

Published online by Cambridge University Press:  07 January 2019

Manfred G. Madritsch  and
Robert F. Tichy
Manfred G. Madritsch
Affiliation:
Université de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France email [email protected]
Robert F. Tichy
Affiliation:
Department for Analysis and Number Theory, Graz University of Technology, A-8010 Graz, Austria email [email protected]
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Abstract

We establish Diophantine inequalities for the fractional parts of generalized polynomials, in particular for sequences $\unicode[STIX]{x1D708}(n)=\lfloor n^{c}\rfloor +n^{k}$ with $c>1$ a non-integral real number and $k\in \mathbb{N}$, as well as for $\unicode[STIX]{x1D708}(p)$ where $p$ runs through all prime numbers. This is related to classical work of Heilbronn and to recent results of Bergelson et al.

MSC classification

Primary: 11J54: Small fractional parts of polynomials and generalizations
Secondary: 11L15: Weyl sums 11L20: Sums over primes
Type
Research Article
Information
Mathematika , Volume 65 , Issue 2 , 2019 , pp. 400 - 435
Copyright
Copyright © University College London 2019 

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