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Frequencies of digits, divergence points, and Schmidt games

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  • Olsen, L.

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.

Suggested Citation

  • Olsen, L., 2009. "Frequencies of digits, divergence points, and Schmidt games," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2222-2232.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:5:p:2222-2232
    DOI: 10.1016/j.chaos.2007.10.010
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