Discrepancy estimates for some linear generalized monomials

Roswitha Hofer, and Olivier Ramaré. "Discrepancy estimates for some linear generalized monomials." Acta Arithmetica 173.2 (2016): 183-196. <http://eudml.org/doc/278856>.

@article{RoswithaHofer2016,
abstract = {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 $_\{α,β,ε\}(N^\{-1+ε\})$.},
author = {Roswitha Hofer, Olivier Ramaré},
journal = {Acta Arithmetica},
keywords = {discrepancy; generalized polynomials; Beatty sequence},
language = {eng},
number = {2},
pages = {183-196},
title = {Discrepancy estimates for some linear generalized monomials},
url = {http://eudml.org/doc/278856},
volume = {173},
year = {2016},
}

TY - JOUR
AU - Roswitha Hofer
AU - Olivier Ramaré
TI - Discrepancy estimates for some linear generalized monomials
JO - Acta Arithmetica
PY - 2016
VL - 173
IS - 2
SP - 183
EP - 196
AB - 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 $_{α,β,ε}(N^{-1+ε})$.
LA - eng
KW - discrepancy; generalized polynomials; Beatty sequence
UR - http://eudml.org/doc/278856
ER -