 

Manash Mukherjee and Gunther Karner
Irrational Numbers of Constant Type  A New Characterization


Published: 
February 21, 1998 
Keywords: 
Irrational numbers, Continued fractions 
Subject: 
11A55 


Abstract
Given an irrational number
α
and a positive integer m,
the distinct fractional parts of α, 2α, ..., mα
determine a partition of the interval
[0,1].
Defining d_{α}(m) and
d'_{α}(m) to be
the maximum and minimum lengths, respectively, of the
subintervals of the
partition corresponding to the integer m,
it is shown that the sequence
(d_{α}(m)/d'_{α}(m))_{m=1}^{∞} is
bounded if and only if α is of constant type.
(The proof of this assertion is based on the
continued fraction expansion of irrational numbers.)


Author information
Manash Mukherjee:
Mathematical Physics Group, Department of Physics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 USA
Current Address: Department of Physics, University of Cincinnati, Cincinnati, Ohio 452210011 USA
manash@physics.uc.edu
Gunther Karner:
Institut für Kerntechnik und Reaktorsicherheit, Universität Karlsruhe (TH), Postfach 3640, D76021 Karlsruhe, Germany
karner@irs.fzk.de

