Divergence points and normal numbers

In Olsen and Winter (J Lond Math Soc 67(2):103-122, 2003) and Baek et al. (Advan Math 214:267-287, 2007) the authors have introduced the notion of "normal" and "non-normal" points of a self-similar set as a main tool for studying the Hausdorff and the packing dimensions of a set of divergence points of self-similar measures. In this paper we will extend the results about the Hausdorff and the packing dimensions of "non-normal" points of a self-similar set in a point of view of Bisbas (Bulletin des Sciences Mathématiques 129(1):25-37, 2005). Namely, we will prove that both the Hausdorff and packing dimensions remain the same if we consider subsets determined by the normality to some bases. This will be proved using the techniques from Bisbas (Bulletin des Sciences Mathématiques 129(1):25-37, 2005) and the construction of suitable measures. Simultaneously this will also give simpler proofs of some of the results from Olsen and Winter (J Lond Math Soc 67(2):103-122, 2003) and Baek et al. (Advan Math 214:267-287, 2007).