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\(1/2\)-Heavy sequences driven by rotation

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Abstract

We investigate the set of \(x \in S^1\) such that for every positive integer \(N\), the first \(N\) points in the orbit of \(x\) under rotation by irrational \(\theta \) contain at least as many values in the interval \([0,1/2]\) as in the complement. By using a renormalization procedure, we show both that the Hausdorff dimension of this set is the same constant (strictly between zero and one) for almost-every \(\theta \), and that for every \(d \in [0,1]\) there is a dense set of \(\theta \) for which the Hausdorff dimension of this set is \(d\).

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Acknowledgments

The author was supported by the Center for Advanced Studies at Ben Gurion University of the Negev as well as the Israeli Council for Higher Education during initial preparation of this manuscript. The author would also like to thank the anonymous referees for valuable suggestions on improving the clarity of presentation, as well as catching several embarrassing typos.

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Authors and Affiliations

  1. Department of Mathematics/CIS, SUNY College at Old Westbury, PO 210, Old Westbury, NY, 11568, USA

    David Ralston

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  1. David Ralston
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Correspondence to David Ralston.

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Communicated by H. Bruin.

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Ralston, D. \(1/2\)-Heavy sequences driven by rotation. Monatsh Math 175, 595–612 (2014). https://doi.org/10.1007/s00605-014-0662-y

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  • DOI: https://doi.org/10.1007/s00605-014-0662-y

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