The multiplicative golden mean shift has infinite Hausdorff measure

Yuval Peres; Boris Solomyak

doi:10.1007/978-0-8176-8400-6_10

The multiplicative golden mean shift has infinite Hausdorff measure

Yuval Peres, Boris Solomyak

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

3 Scopus citations

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_κ) satisfies x_κx_2κ = 0 for all _κ ≥ 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.

Original language	English
Title of host publication	Further Developments in Fractals and Related Fields
Editors	Julien Barral, Stéphane Seuret
Publisher	Springer International Publishing
Pages	193-212
Number of pages	20
ISBN (Print)	9780817683993
DOIs	https://doi.org/10.1007/978-0-8176-8400-6_10
State	Published - 2013
Externally published	Yes
Event	International Conference on Fractals and Related Fields, 2011 - Porquerolles Island, France Duration: 1 Jun 2011 → …

Publication series

Name	Trends in Mathematics
Volume	55
ISSN (Print)	2297-0215
ISSN (Electronic)	2297-024X

Conference

Conference	International Conference on Fractals and Related Fields, 2011
Country/Territory	France
City	Porquerolles Island
Period	1/06/11 → …

Bibliographical note

Publisher Copyright:
© Springer Science+Business Media New York 2013.

Access to Document

10.1007/978-0-8176-8400-6_10

Cite this

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title = "The multiplicative golden mean shift has infinite Hausdorff measure",

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 (xκ) satisfies xκx2κ = 0 for all κ ≥ 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.",

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Peres, Y & Solomyak, B 2013, The multiplicative golden mean shift has infinite Hausdorff measure. in J Barral & S Seuret (eds), Further Developments in Fractals and Related Fields. Trends in Mathematics, vol. 55, Springer International Publishing, pp. 193-212, International Conference on Fractals and Related Fields, 2011, Porquerolles Island, France, 1/06/11. https://doi.org/10.1007/978-0-8176-8400-6_10

The multiplicative golden mean shift has infinite Hausdorff measure. / Peres, Yuval; Solomyak, Boris.
Further Developments in Fractals and Related Fields. ed. / Julien Barral; Stéphane Seuret. Springer International Publishing, 2013. p. 193-212 (Trends in Mathematics; Vol. 55).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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The multiplicative golden mean shift has infinite Hausdorff measure

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