Discrepancy estimates for some linear generalized monomials

Volume 173 / 2016

Roswitha Hofer, Olivier Ramaré Acta Arithmetica 173 (2016), 183-196 MSC: Primary 11K38; Secondary 11K31, 11J82. DOI: 10.4064/aa8164-12-2015 Published online: 25 March 2016


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 $\mathcal{O}_{\alpha,\beta,\varepsilon}(N^{-1+\varepsilon})$.


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

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