Digit frequencies and Bernoulli convolutions

Thomas Michael William Kempton

doi:10.1016/j.indag.2014.04.011

Digit frequencies and Bernoulli convolutions

Thomas Michael William Kempton

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

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Abstract

It is well known that when β is a Pisot number, the corresponding Bernoulli convolution ν(β) has Hausdorff dimension less than 1, i.e. that there exists a set A(β) with (ν(β))(A(β))=1 and dim_H(A(β))<1. We show explicitly how to construct for each Pisot number β such a set A(β).

Original language	English
Pages (from-to)	832-842
Journal	Indagationes Mathematicae
Volume	25
Issue number	4
DOIs	https://doi.org/10.1016/j.indag.2014.04.011
Publication status	Published - 27 Jun 2014

Keywords

Bernoulli convolutions
Beta expansions
Ergodic theory

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10.1016/j.indag.2014.04.011

Digit Frequencies and Bernoulli Convolutions
© 2014, Publisher / the Author(s). This work is made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at www.sciencedirect.com / https://dx.doi.org/10.1016/j.indag.2014.04.011
Accepted author manuscript, 160 KB

Non-conformal repellers: Fractal and multifractal structure of non-conformal repellers
Falconer, K. J. (PI)
EPSRC
13/01/14 → 12/01/17
Project: Standard

Cite this

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title = "Digit frequencies and Bernoulli convolutions",

abstract = "It is well known that when β is a Pisot number, the corresponding Bernoulli convolution ν(β) has Hausdorff dimension less than 1, i.e. that there exists a set A(β) with (ν(β))(A(β))=1 and dim_H(A(β))<1. We show explicitly how to construct for each Pisot number β such a set A(β).",

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author = "Kempton, {Thomas Michael William}",

note = "This work was supported partly by the Dutch Organisation for Scientific Research (NWO) grant number 613.001.022 and partly by the Engineering and Physical Sciences Research Council grant number EP/K029061/1. ",

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Y1 - 2014/6/27

N2 - It is well known that when β is a Pisot number, the corresponding Bernoulli convolution ν(β) has Hausdorff dimension less than 1, i.e. that there exists a set A(β) with (ν(β))(A(β))=1 and dim_H(A(β))<1. We show explicitly how to construct for each Pisot number β such a set A(β).

AB - It is well known that when β is a Pisot number, the corresponding Bernoulli convolution ν(β) has Hausdorff dimension less than 1, i.e. that there exists a set A(β) with (ν(β))(A(β))=1 and dim_H(A(β))<1. We show explicitly how to construct for each Pisot number β such a set A(β).

KW - Bernoulli convolutions

KW - Beta expansions

KW - Ergodic theory

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JO - Indagationes Mathematicae

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Digit frequencies and Bernoulli convolutions

Abstract

Keywords

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Non-conformal repellers: Fractal and multifractal structure of non-conformal repellers

Cite this