Applications of multifractal divergence points to some sets of d-tuples of numbers defined by their N-adic expansion

Abstract

In this paper we apply the techniques and results from the theory of multifractal divergence points developed in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591–1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103–122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] to give a systematic and detailed account of the Hausdorff dimensions of sets of d-tuples numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using the method and results from [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, Journal de Mathématiques Pures et Appliquées 82 (2003) 1591–1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages III, Preprint (2002); L. Olsen, S. Winter, J. London Math. Soc. 67 (2003) 103–122; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages II, Preprint (2001)] we investigate and compute the Hausdorff dimension of several new sets of d-tuples of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of d-tuples numbers for which the limiting frequencies of the digits in their N-adic expansion do not exist. Such sets have only very rarely been studied. In addition, our techniques provide simple proofs of higher-dimensional and non-linear generalizations of known results, by Cajar and Volkmann and others, on the Hausdorff dimension of sets of normal and non-normal numbers.