Given an irrational number $\alpha$ and a positive integer $m$, the distinct fractional parts of $\alpha, 2\alpha, \cdots, m\alpha$ determine a partition of the interval $[0,1]$. Defining $\displaystyle{d_{\alpha}(m)}$ and $\displaystyle{d'_{\alpha}(m)}$ 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 $\displaystyle{\left(\frac{d_{\alpha}(m)} {d'_{\alpha}(m)} \right)_{m=1}^{\infty}}$ is bounded if and only if $\alpha$ is of constant type. (The proof of this assertion is based on the continued fraction expansion of irrational numbers.)