On sets of fractional dimensions (II)

Abstract

1. Any number x between 0 and 1 may be expressed uniquely in the formwhere xr is a non-negative integer less than r (r = 2,3,…). We consider the set E of numbers x for whichWe establish an inequality connecting the dimensional number of the set E with certain constants of the seriesin particular we show that, when ξτ = rθ, the dimensional number of E is θ. We are concerned with the measure of Hausdorff.