Summability of Sequences of 0'S and 1'S

Abstract

In general, if tn is defined for each n and tends to s as n -oo we say that the sequence {Sk} is summable T to the value s. Let X denote the class of all sequences x ak) where the axe are 0 or 1, with infinitely many of them having the value 1. This restriction is imposed in order that the mapping of X into D, which is given below, will be one-to-one; it does not affect the results. The problems treated in this paper have their origin in the theorem of Borel mentioned in the next paragraph, and in the following theorem of Steinhaus [7]'. (1.6) THEOREM. Corresponding to each T there is at least one x E X which is not summable T. We shall denote the set of all such x by the symbol Xo(T) and its complement with respect to I by X1(T). The set X1(T) is therefore composed of all x E X which are summable T. The subset X2(T) of X1(T), consisting of all x which are summable T to the value 4, will be of particular interest. We now observe that a one-to-one mapping of X into the interval D =(0 < y < 1) may be obtained by defining y as the dyadic fraction 0. aOala2 ... if x = (ao, ai., a2, ... ) is a point of X, and conversely. The notation Di (T) for i = 0, 1, 2, will be used for the set of y corresponding to 3t(T) by means of this mapping. By the