The representation of real numbers
Ottis W. Rechard · Proceedings of the American Mathematical Society · 1950
where f(x) is an arbitrary function in the class E,. Some functions (for example, f(x) =x/p, which leads to the representation of a number as a decimal to the base p) when employed in the algorithm (A) yield one-one correspondences between real numbers and sequences of integers mod p. On the other hand, any function, for example, whose graph has more than one point in common with any of the straight line segments connecting the points (j, 0) and (j+1, 1), j=0, 1, * * *, p-I, will obviously lead to a correspondence which is many-one. We shall denote by Ep* the subclass of Ep consisting of those functions which in the algorithm (A) give rise to one-one correspondences. The present paper contains very simple characterizations of those correspondences between real numbers and sequences of integers mod p which can be obtained by applying the algorithm (A) with functions from the classes E * and Ep -E * respectively. By means of these characterizations it is possible to settle two of the problems raised by Everett and to give an answer (albeit not a completely satisfactory one) to a third, namely that of characterizing the class Ep* itself.