DESCRIPTIONS OF THE CHARACTERISTIC SEQUENCE OF AN IRRATIONAL

Let alpha be a positive irrational real number. (Without loss of generality assume 0 < alpha < 1.) The characteristic sequence of alpha is f(alpha) = f1f2..., where f(n) = [(n + 1)alpha] - [nalpha]. We make some observations on the various descriptions of the characteristic sequence of ce 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 [nalpha] which appears in A. S. Fraenkel, J. Levitt, and M. Shimshoni [17]. Some concluding remarks give conditions on n which are equivalent to f(n) = 1.