<abstract xmlns="http://www.w3.org/1999/xhtml">In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>Z</mi></mrow><mrow><msub><mrow><mi>B</mi></mrow><mn>2</mn></msub><mo>,</mo><mi>ν</mi></mrow><mrow><mi>κ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup></mrow></math><tex-math>Z_{{{\rm B}_2},\nu }^{\kappa ,\mu }</tex-math></alternatives></inline-formula> of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2; α; γ)-diaphony of the nets <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>Z</mi></mrow><mrow><msub><mrow><mi>B</mi></mrow><mn>2</mn></msub><mo>,</mo><mi>ν</mi></mrow><mrow><mi>κ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup></mrow></math><tex-math>Z_{{{\rm B}_2},\nu }^{\kappa ,\mu }</tex-math></alternatives></inline-formula> is shown. If α1 = α2, then the following holds: if 1 &lt; α2 &lt; 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0, if α2 = 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mi>N</mi></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over N}</tex-math></alternatives></inline-formula>) and if α2 > 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_005.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0. If α1 > α2, then the following holds: if 1 &lt; α2 &lt; 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_006.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0, if α2 = 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_007.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac></mrow></math><tex-math>{1 \over N}</tex-math></alternatives></inline-formula>) and if α2 > 2 the exact order is 𝒪 (<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_008.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msqrt><mrow><mo>log</mo><mi>N</mi></mrow></msqrt></mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>ε</mi></mrow></msup></mrow></mfrac></mrow></math><tex-math>{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}</tex-math></alternatives></inline-formula>) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0002_eq_009.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>Z</mi></mrow><mrow><msub><mrow><mi>B</mi></mrow><mn>2</mn></msub><mo>,</mo><mi>ν</mi></mrow><mrow><mi>κ</mi><mo>,</mo><mi>μ</mi></mrow></msubsup></mrow></math><tex-math>Z_{{{\rm B}_2},\nu }^{\kappa ,\mu }</tex-math></alternatives></inline-formula>.</abstract>

In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2; α; γ)-diaphony of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } is shown. If α1 = α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α2 = 2 the exact order is 𝒪 (logNN{{\sqrt {\log N} } \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. If α1 > α2, then the following holds: if 1 < α2 < 2 the exact order is 𝒪 (logNN1-ε{{\sqrt {\log N} } \over {{N^{1 - \varepsilon }}}}) for some ε > 0, if α2 = 2 the exact order is 𝒪 (1N{1 \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu }.

On the (Vil<italic>B</italic>2; <italic>α</italic>; <italic>γ</italic>)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

South West University “Neofit Rilski”

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the (Vil; ; )-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of two-dimensional nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } of type of Zaremba-Halton constructed in a generalized B2-adic system or Cantor system is introduced and the (VilB2; α; γ)-diaphony of these nets is studied. The influence of the vector α = (α1, α2) of exponential parameters to the exact order of the (VilB2; α; γ)-diaphony of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu } is shown. If α1 = α2, then the following holds: if 1  0, if α2 = 2 the exact order is 𝒪 (logNN{{\sqrt {\log N} } \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. If α1 > α2, then the following holds: if 1  0, if α2 = 2 the exact order is 𝒪 (1N{1 \over N}) and if α2 > 2 the exact order is 𝒪 (logNN1+ε{{\sqrt {\log N} } \over {{N^{1 + \varepsilon }}}}) for some ε > 0. Here N = Bν, where Bν denotes the number of the points of the nets ZB2,νκ,μZ_{{{\rm B}_2},\nu }^{\kappa ,\mu }.

In the present paper the so-called (VilBs; α; γ)-diaphony as a quantitative measure for the distribution of sequences and nets is considered. A class of...

On the (Vil_B₂; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Published Online: Jul 24, 2020

Page range: 27 - 50

Received: Dec 20, 2019

Accepted: Mar 03, 2020

DOI: https://doi.org/10.2478/udt-2020-0002

Keywords
Diaphony, Vilenkin function, Walsh function, nets of type of Zaremba-Halton, van der Corput sequence

© 2020 Vesna Dimitrievska Ristovska et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the (VilB2; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Published Online: Jul 24, 2020

Page range: 27 - 50

Received: Dec 20, 2019

Accepted: Mar 03, 2020

DOI: https://doi.org/10.2478/udt-2020-0002

KeywordsDiaphony, Vilenkin function, Walsh function, nets of type of Zaremba-Halton, van der Corput sequence

© 2020 Vesna Dimitrievska Ristovska et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

On the (Vil_B₂; α; γ)-Diaphony of the Nets of Type of Zaremba–Halton Constructed in Generalized Number System

Keywords
Diaphony, Vilenkin function, Walsh function, nets of type of Zaremba-Halton, van der Corput sequence