Discrepancy estimates for some linear generalized monomials
Volume 173 / 2016
Abstract
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 $\mathcal{O}_{\alpha,\beta,\varepsilon}(N^{-1+\varepsilon})$.
