The Multiplicative Golden Mean Shift Has Infinite Hausdorff Measure
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. 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.
KeywordsProbability Measure Hausdorff Dimension Binary Sequence Hausdorff Measure Symbolic Space
The research of B. S. was supported in part by the NSF grant DMS-0968879. He is grateful to the Microsoft Research Theory Group for hospitality during 2010–2011. He would also like to thank the organizers of the conference “Fractals and Related Fields II” for the excellent meeting and stimulating atmosphere. The authors are grateful to the referee for careful reading of the manuscript and many helpful comments.
- 1.Bedford, T.: Crinkly curves, Markov partitions and box dimension in self-similar sets. Ph.D. Thesis, University of Warwick, (1984)Google Scholar
- 3.Fan, A., Liao, L., Ma, J.: Level sets of multiple ergodic averages. Preprint (2011) arXiv:1105.3032. Monatsh. Math. 168(1), 17–26 (2012)Google Scholar
- 6.Kenyon, R., Peres, Y., Solomyak, B.: Hausdorff dimension for fractals invariant under the multiplicative integers. Preprint (2011) arXiv 1102.5136. Ergodic Th. Dynam. Sys. 32(5), 1567–1584 (2012)Google Scholar