Bounded Lüroth expansions: applying Schmidt games where infinite distortion exists

Tom 158 / 2013

Bill Mance, Jimmy Tseng Acta Arithmetica 158 (2013), 33-47 MSC: Primary 11K55; Secondary 11K50. DOI: 10.4064/aa158-1-2


We show that the set of numbers with bounded Lüroth expansions (or bounded Lüroth series) is winning and strong winning. From either winning property, it immediately follows that the set is dense, has full Hausdorff dimension, and satisfies a countable intersection property. Our result matches the well-known analogous result for bounded continued fraction expansions or, equivalently, badly approximable numbers.

We note that Lüroth expansions have a countably infinite Markov partition, which leads to the notion of infinite distortion (in the sense of Markov partitions).


  • Bill ManceDepartment of Mathematics
    Ohio State University
    Columbus, OH 43210, U.S.A.
  • Jimmy TsengDepartment of Mathematics
    University of Illinois at Urbana-Champaign
    Urbana, IL 61801, U.S.A.

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