New York Journal of Mathematics
Volume 4 (1998) 31-34


Manash Mukherjee and Gunther Karner

Irrational Numbers of Constant Type -- A New Characterization

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

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 dα(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 (dα(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 45221-0011 USA

Gunther Karner:
Institut für Kerntechnik und Reaktorsicherheit, Universität Karlsruhe (TH), Postfach 3640, D-76021 Karlsruhe, Germany