Non-normal numbers to different bases and their Hausdorff dimension (Q1081636)

The paper gives an exposition on recent metric results concerning digit properties of real numbers relative to different bases. In particular, a new, relatively simple proof of a theorem of \textit{A. D. Pollington} [Pac. J. Math. 95, 193-204 (1981; Zbl 0479.10031)] is outlined. The following theorem is among the new results which are stated: Let C be the set of all \(x\in [0,1]\) whose base s expansion only involves the digits 0,1,...,t; \(t<s\), and let D be the set of all \(x\in [0,1]\) which are not simply normal to the base \(s^ n\), \(n=1,2,...\), but normal to others. Then the sets \(C\cap D\) and C have the same Hausdorff dimension.