TY - JOUR

T1 - Applications of multifractal divergence points to sets of numbers defined by their $N$-adic expansion

AU - Olsen, Lars Ole Ronnow

PY - 2004/1

Y1 - 2004/1

N2 - In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of 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 (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

AB - In this paper we apply the techniques and results from the theory of multifractal divergence points to give a systematic and detailed account of the Hausdorff dimensions of sets of numbers defined in terms of the asymptotic behaviour of the frequencies of the digits in their N-adic expansion. Using earlier methods and results we investigate and compute the Hausdorff dimension of several new sets of numbers. In particular, we compute the Hausdorff dimension of a large class of sets of 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 (substantial generalizations of) known results, by Cajar and Drobot and Turner and others, on the Hausdorff dimension of sets of normal and non-normal numbers.

KW - PERRON-FROBENIUS THEORY

KW - HAUSDORFF DIMENSION

UR - http://www.scopus.com/inward/record.url?scp=1042279260&partnerID=8YFLogxK

U2 - 10.1017/S0305004103007047

DO - 10.1017/S0305004103007047

M3 - Article

SN - 0305-0041

VL - 136

SP - 139

EP - 165

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

IS - 1

ER -