The Multiplicative golden mean shift has infinite Hausdorff measure

2012/01/27 by Yuval Peres, Peres, Yuval, Boris Solomyak +1
Computer Science · Mathematics · Physics and Astronomy · #37C45 (Primary) 28A78 #37D35 (Secondary) #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1201.5842

openalex publication_date 2012/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In an earlier work, joint with R. Kenyon, we computed the Hausdorff dimension of the "multiplicative golden mean shift" defined as the set of all reals in [0,1] whose binary expansion (x_k) satisfies x_k x_2k=0 for all k=1,2... Here we show that this set has infinite Hausdorff measure in its dimension. A more precise result in terms of gauges in which the Hausdorff measure is infinite is also obtained.

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