Irrational Numbers of Constant Type — A New Characterization

Abstract

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 dα(m) and d ′ α(m) to be the maximum and minimum lengths, respectively, of the subintervals of�the partition corresponding to the integer dα(m) m, it is shown that the sequence d ′ α (m) � ∞ is bounded if and only if α m=1 is of constant type. (The proof of this assertion is based on the continue