Frequencies of digits, divergence points, and Schmidt games

L. Olsen

Chaos, Solitons & Fractals, 2009, vol. 40, issue 5, 2222-2232

Abstract: Sets of divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequency of a given string of N-adic digits of x fails to exist, have recently attracted huge interest in the literature. In this paper we consider sets of simultaneous divergence points, i.e. numbers x (or tuples of numbers) for which the limiting frequencies of all strings of N-adic digits of x fail to exist. We show that many natural sets of simultaneous divergence points are (α,β)-wining sets in the sense of the Schmidt game. As an application we obtain lower bounds for the Hausdorff dimension of these sets.

Date: 2009
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:40:y:2009:i:5:p:2222-2232

DOI: 10.1016/j.chaos.2007.10.010

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