Descriptions of the Characteristic Sequence of an Irrational

Let α be a positive irrational real number. (Without loss of generality assume 0 < α < 1.) The characteristic sequence of α is f(α) =f1f2 ···, where fn = [(n + 1)α] - [nα]. We make some observations on the various descriptions of the characteristic sequence of α which have appeared in the literature. We then refine one of these descriptions in order to obtain a very simple derivation of an arithmetic expression for [nα] which appears in A. S. Fraenkel, J. Levitt, and M. Shimshoni [17]. Some concluding remarks give conditions on n which are equivalent to fn = 1.