The Vitali covering principle is a powerful method in a wide class of problems of the theory of functions of a real variable and of the theory of sets.
in Cambridge (England).w The problem of this paper is connected with the well-known Hardy-Litflewood problem on the sum of digits of real numbers represented in the dyadic system1). Denote by P(x,,n) the sum of the first n figures of the number x(0 @, m~E = ~ for any r<@.We say that ~ is the dimensional number of the set E and we write d E = ~ or we express the same by saying that B is a ~o-d/menv/onal set or, more simply, a ~-set.
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