Discrepancy estimates for some linear generalized monomials

• Autorzy: Roswitha Hofer , Olivier Ramaré
• Języki publikacji: EN

Abstrakty

EN

We consider sequences modulo one that are generated using a generalized polynomial over the real numbers. Such polynomials may also involve the integer part operation [·] additionally to addition and multiplication. A well studied example is the (nα) sequence defined by the monomial αx. Their most basic sister, $([nα]β)_{n≥0}$, 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 (α,β) of real numbers is in a certain sense badly approximable, then the discrepancy satisfies a bound of order $𝓞_{α,β,ε}(N^{-1+ε})$.

Kategorie tematyczne

• 11K38: Irregularities of distribution, discrepancy
• 11J82: Measures of irrationality and of transcendence
• 11K31: Special sequences

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2016
• Tom: 173
• Numer: 2
• Strony: 183-196

Daty

wydano

2016

Twórcy

autor

Roswitha Hofer

• Institute of Financial Mathematics, and Applied Number Theory, Johannes Kepler University Linz, Altenbergerstr. 69, A-4040 Linz, Austria

autor

Olivier Ramaré

• CNRS Laboratoire Paul Painlevé, Université Lille I, U.M.R. 8524, 59 655 Villeneuve d'Ascq Cedex, France

Identyfikatory

DOI

10.4064/aa8164-12-2015