Non-L1 functions with rotation sets of Hausdorff dimension one

  title={Non-L1 functions with rotation sets of Hausdorff dimension one},
  author={Zolt'an Buczolich},
  journal={Acta Mathematica Hungarica},
  • Z. Buczolich
  • Published 2010
  • Mathematics
  • Acta Mathematica Hungarica
Suppose that f: ℝ → ℝ is a given measurable function, periodic by 1. For an α ∈ ℝ put Mnαf(x) = 1/n+1 Σk=0nf(x + kα). Let Γf denote the set of those α’s in (0;1) for which Mnαf(x) converges for almost every x ∈ ℝ. We call Γf the rotation set of f. We proved earlier that from |Γf| > 0 it follows that f is integrable on [0; 1], and hence, by Birkhoff’s Ergodic Theorem all α ∈ [0; 1] belongs to Γf. However, Γf\ℚ can be dense (even c-dense) for non-L1 functions as well. In this paper we show that… 

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