2011/06/03 by David Ralston, Ralston, David
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT
paper · pdf · doi:10.48550/arxiv.1106.0577
in review
arxiv created 2011/06/03 · openalex publication_date 2011/06/03 · arxiv updated 2011/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We investigate the set of x ∈ S1 such that for every positive integer N, the first N points in the orbit of x under rotation by irrational θ 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 θ, and that for every d ∈ [0,1] there is a dense set of θ for which the Hausdorff dimension of this set is d.