<abstract xmlns="http://www.w3.org/1999/xhtml">The Bernoulli convolution associated to the real <italic>β</italic> > 1 and the probability vector (<italic>p</italic>0, . . . , <italic>pd</italic>−1) is a probability measure <italic>ηβ,p</italic> on ℝ, solution of the self-similarity relation <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mi>η</mi><mo>=</mo><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><msub><mi>p</mi><mi>k</mi></msub><mo>⋅</mo><mi>η</mi><mi>○</mi><msubsup><mi>S</mi><mi>k</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup></mrow></mstyle></math><tex-math>$\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $</tex-math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0015_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>S</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>x</mi><mo>+</mo><mi>k</mi></mrow><mi>β</mi></mfrac></math><tex-math>$S_k (x) = {{x + k} \over \beta }$</tex-math></alternatives></inline-formula>. If <italic>β</italic> is an integer or a Pisot algebraic number with finite Rényi expansion, <italic>ηβ,p</italic> is sofic and a Markov chain is naturally associated. If <italic>β</italic> = <italic>b</italic> ∈ ℕ and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0015_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>p</mi><mn>0</mn></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mi>p</mi><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>d</mi></mfrac></math><tex-math>$p_0 = \cdots = p_{d - 1} = {1 \over d}$</tex-math></alternatives></inline-formula>, the study of <italic>ηb,p</italic> is close to the study of the order of growth of the number of representations in base <italic>b</italic> with digits in {0, 1, . . . , <italic>d</italic> − 1}. In the case <italic>b</italic> = 2 and <italic>d</italic> = 3 it has also something to do with the metric properties of the continued fractions.</abstract>

The Bernoulli convolution associated to the real β > 1 and the probability vector (p0, . . . , pd−1) is a probability measure ηβ,p on ℝ, solution of the self-similarity relation η=∑k=0d−1pk⋅η○Sk−1$\eta = \sum\nolimits_{k = 0}^{d - 1} {p_k \cdot \eta \circ S_k^{ - 1} } $, where Sk(x)=x+kβ$S_k (x) = {{x + k} \over \beta }$. If β is an integer or a Pisot algebraic number with finite Rényi expansion, ηβ,p is sofic and a Markov chain is naturally associated. If β = b ∈ ℕ and p0=⋯=pd−1=1d$p_0 = \cdots = p_{d - 1} = {1 \over d}$, the study of ηb,p is close to the study of the order of growth of the number of representations in base b with digits in {0, 1, . . . , d − 1}. In the case b = 2 and d = 3 it has also something to do with the metric properties of the continued fractions.

Sofic Measures and Densities of Level Sets

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Uniform distribution theory

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{"article-title":"Sofic Measures and Densities of Level Sets"}

The Bernoulli convolution associated to the real β > 1 and the probability vector (p0, . . . , pd−1) is a probability measure ηβ,p on ℝ, solution of the...

Sofic Measures and Densities of Level Sets

Published Online: Jan 13, 2017

Page range: 91 - 124

Received: Nov 09, 2014

Accepted: Oct 11, 2015

DOI: https://doi.org/10.1515/udt-2016-0015

Keywords
partition function, numeration system, radix expansion, Pisot scale, Bernoulli convolutions

© 2016 Alain Thomas, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Sofic Measures and Densities of Level Sets

Published Online: Jan 13, 2017

Page range: 91 - 124

Received: Nov 09, 2014

Accepted: Oct 11, 2015

DOI: https://doi.org/10.1515/udt-2016-0015

Keywordspartition function, numeration system, radix expansion, Pisot scale, Bernoulli convolutions

© 2016 Alain Thomas, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
partition function, numeration system, radix expansion, Pisot scale, Bernoulli convolutions