Discrepancy estimates for some linear generalized monomials

Roswitha Hofer ¹   ; Olivier Ramaré ²

¹ Institute of Financial Mathematics and Applied Number Theory Johannes Kepler University Linz Altenbergerstr. 69 A-4040 Linz, Austria
² CNRS Laboratoire Paul Painlevé Université Lille I U.M.R. 8524 59 655 Villeneuve d’Ascq Cedex, France

Acta Arithmetica, Tome 173 (2016) no. 2, pp. 183-196

Résumé

We consider sequences modulo one that are generated using a \emph{generalized} polynomial over the real numbers. Such polynomials may also involve the integer part operation $[\cdot]$ additionally to addition and multiplication. A well studied example is the $(n \alpha)$ sequence defined by the monomial $\alpha x$. Their most basic sister, $([n \alpha]\beta)_{n\geq 0}$, is less investigated. So far only the uniform distribution modulo one of these sequences is resolved. Completely new, however, are the discrepancy results proved in this paper. We show in particular that if the pair $(\alpha,\beta)$ of real numbers is in a certain sense badly approximable, then the discrepancy satisfies a bound of order $\mathcal{O}_{\alpha,\beta,\varepsilon}(N^{-1+\varepsilon})$.

Détail

DOI : 10.4064/aa8164-12-2015

Keywords: consider sequences modulo generated using emph generalized polynomial real numbers polynomials may involve integer part operation cdot additionally addition multiplication studied example alpha sequence defined monomial alpha their basic sister alpha beta geq investigated far only uniform distribution modulo these sequences resolved completely however discrepancy results proved paper particular pair alpha beta real numbers certain sense badly approximable discrepancy satisfies bound order mathcal alpha beta varepsilon varepsilon