<abstract xmlns="http://www.w3.org/1999/xhtml">For a set <italic>A</italic> of positive integers <italic>a</italic>1<italic>&lt; a</italic>2<italic>&lt; · · ·</italic>, let <italic><underline>d</underline></italic>(<italic>A</italic>), <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>d</mi><mo>¯</mo></mover><mo>(</mo><mi>A</mi><mo>)</mo></math><tex-math>$\overline d (A)$</tex-math></alternatives></inline-formula> denote its lower and upper asymptotic densities. The gap density is defined as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mtext>lim</mtext><mo>⁡</mo><mtext> </mtext><msub><mtext>sup</mtext><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><mfrac><mrow><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mrow><msub><mi>a</mi><mi>n</mi></msub></mrow></mfrac></math><tex-math>$\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$</tex-math></alternatives></inline-formula>. The paper investigates the class 𝒢(<italic>α, β, γ</italic>) of all sets <italic>A</italic> with <italic><underline>d</underline></italic>(<italic>A</italic>) = <italic>α</italic>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mover><mi>d</mi><mo>¯</mo></mover><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mi>β</mi></math><tex-math>$\overline d (A) = \beta $</tex-math></alternatives></inline-formula> and <italic>λ</italic>(<italic>A</italic>) = <italic>γ</italic> for given <italic>α, β, γ</italic> with 0 <italic>≤ α ≤ β ≤</italic> 1 <italic>≤ γ</italic> and <italic>αγ ≤ β</italic>. Using the classical dyadic mapping <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>ϱ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mstyle><msubsup><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mi>∞</mi></msubsup><mrow><mfrac><mrow><msub><mi>χ</mi><mi>A</mi></msub><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mn>2</mn><mi>n</mi></msup></mrow></mfrac></mrow></mstyle></math><tex-math>$\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $</tex-math></alternatives></inline-formula>, where <italic>χA</italic> is the characteristic function of <italic>A</italic>, the main result of the paper states that the ϱ-image set ϱ<italic>𝒢</italic>(<italic>α, β, γ</italic>) has the Hausdorff dimension
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_005.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>dim</mi><mo>ϱ</mo><mi>𝒢</mi><mo>(</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo>⁡</mo><mrow><mo>{</mo><mrow><mi>δ</mi><mo>(</mo><mi>α</mi><mo>)</mo><mo>,</mo><mi>δ</mi><mo>(</mo><mi>β</mi><mo>)</mo><mo>,</mo><mfrac><mn>1</mn><mi>γ</mi></mfrac><munder><mrow><mi>max</mi><mo>⁡</mo></mrow><mrow><mi>σ</mi><mo>∈</mo><mo>[</mo><mi>α</mi><mi>γ</mi><mo>,</mo><mi>β</mi><mo>]</mo></mrow></munder><mi>δ</mi><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mo>}</mo></mrow><mo>,</mo></math><tex-math>$$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$</tex-math></alternatives></disp-formula>
where <italic>δ</italic> is the entropy function
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2016-0007_eq_006.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mi>δ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>−</mo><mi>x</mi><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mi>x</mi><mo>−</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo><mo> </mo><msub><mi>log</mi><mn>2</mn></msub><mo> </mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo>)</mo><mo>.</mo></math><tex-math>$$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$</tex-math></alternatives></disp-formula></abstract>

For a set A of positive integers a1< a2< · · ·, let d(A), d¯(A)$\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as λ(A)=lim⁡ supn→∞an+1an$\lambda (A) = \lim \;{\rm sup} _{n \to \infty } {{a_{n + 1} } \over {a_n }}$. The paper investigates the class 𝒢(α, β, γ) of all sets A with d(A) = α, d¯(A)=β$\overline d (A) = \beta $ and λ(A) = γ for given α, β, γ with 0 ≤ α ≤ β ≤ 1 ≤ γ and αγ ≤ β. Using the classical dyadic mapping ϱ(A)=∑n=1∞χA(n)2n$\varrho (A) = \sum\nolimits_{n = 1}^\infty {{{\chi _A (n)} \over {2^n }}} $, where χA is the characteristic function of A, the main result of the paper states that the ϱ-image set ϱ𝒢(α, β, γ) has the Hausdorff dimension
dimϱ𝒢(α,β,γ)=min⁡{δ(α),δ(β),1γmax⁡σ∈[αγ,β]δ(σ)},$$\dim \varrho \cal {G}(\alpha ,\beta ,\gamma ) = \min \left\{ {\delta (\alpha ),\delta (\beta ), { 1 \over \gamma }\mathop {\max }\limits_{\sigma \in [\alpha \gamma ,\beta ]} \delta (\sigma )} \right\},$$
where δ is the entropy function
δ(x)=−x log2 x−(1−x) log2 (1−x).$$\delta (x) = - x\log _2 x - (1 - x)\;\log _2 (1 - x).$$

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

Department of Mathematics, University of Ostrava & Centre of Excellence IT4Innovations-Division UO-IRAFM, 30. dubna 22, 701 03 Ostrava 1,

Department of Mathematics, University of Ostrava, 30. dubna 22, Ostrava, 701 03,

Uniform distribution theory

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

{"article-title":"On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities"}

For a set A of positive integers a1< a2< · · ·, let d(A), d¯(A)$\overline d (A)$ denote its lower and upper asymptotic densities. The gap density is defined as...

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

Published Online: Jan 13, 2017

Page range: 141 - 157

Received: Jul 20, 2012

Accepted: Apr 10, 2015

DOI: https://doi.org/10.1515/udt-2016-0007

Keywords
Sequences of integers, lower asymptotic density, upper asymptotic density, gap density, Hausdorff dimension

© 2016 Ladislav Mišík et al., published by De Gruyter Open

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

On Hausdorff Dimensions Related to Sets with Given Asymptotic and Gap Densities

Published Online: Jan 13, 2017

Page range: 141 - 157

Received: Jul 20, 2012

Accepted: Apr 10, 2015

DOI: https://doi.org/10.1515/udt-2016-0007

KeywordsSequences of integers, lower asymptotic density, upper asymptotic density, gap density, Hausdorff dimension

© 2016 Ladislav Mišík et al., published by De Gruyter Open

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

Keywords
Sequences of integers, lower asymptotic density, upper asymptotic density, gap density, Hausdorff dimension