<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>q</italic> be an integer greater than or equal to 2, and let <italic>S</italic><italic>q</italic>(<italic>n</italic>)denote the sum of digits of <italic>n</italic> in base <italic>q</italic>.For<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mrow><mo>[</mo><mrow><mn>0</mn><mo>;</mo><mover><mrow><mn>1</mn><mo>,</mo><mi>m</mi></mrow><mo>¯</mo></mover></mrow><mo>]</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo></mrow></math><tex-math>\alpha = \left[ {0;\overline {1,m} } \right],\,\,\,m \ge 2,</tex-math></alternatives></disp-formula>let <italic>S</italic>α(<italic>n</italic>) denote the sum of digits in the Ostrowski <italic>α</italic>-representation of <italic>n</italic>. Let <italic>m</italic>1,<italic>m</italic>2 ≥ 2 be integers with<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>gcd</mo><mrow><mo>(</mo><mrow><mi>q</mi><mo>-</mo><mn>1</mn><mo>,</mo><msub><mrow><mi>m</mi></mrow><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mo>gcd</mo><mrow><mo>(</mo><mrow><mi>m</mi><mo>,</mo><msub><mrow><mi>m</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math><tex-math>\gcd \left( {q - 1,{m_1}} \right) = \gcd \left( {m,{m_2}} \right) = 1</tex-math></alternatives></disp-formula>We prove that there exists <italic>δ></italic> 0 such that for all integers <italic>r</italic>1,<italic>r</italic>2,<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr><mtd><mrow><mrow><mo>|</mo><mrow><mrow><mo>{</mo><mrow><mn>0</mn><mo>≤</mo><mi>n</mi><mo>&lt;</mo><mi>N</mi><mo>:</mo><msub><mrow><mi>S</mi></mrow><mi>q</mi></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≡</mo><msub><mrow><mi>r</mi></mrow><mn>1</mn></msub><mrow><mo>(</mo><mrow><mo>mod</mo><mi> </mi><msub><mrow><mi>m</mi></mrow><mn>1</mn></msub></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><msub><mrow><mi>S</mi></mrow><mi>α</mi></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>≡</mo><msub><mrow><mi>r</mi></mrow><mn>2</mn></msub><mrow><mo>(</mo><mrow><mo>mod</mo><mi> </mi><msub><mrow><mi>m</mi></mrow><mn>2</mn></msub></mrow><mo>)</mo></mrow></mrow><mo>}</mo></mrow></mrow><mo>|</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mo>=</mo><mfrac><mi>N</mi><mrow><msub><mrow><mi>m</mi></mrow><mn>1</mn></msub><msub><mrow><mi>m</mi></mrow><mn>2</mn></msub></mrow></mfrac><mo>+</mo><mn>0</mn><mrow><mo>(</mo><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>δ</mi></mrow></msup></mrow><mo>)</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></mrow></math><tex-math>\matrix{ {\left| {\left\{ {0 \le n &lt; N:{S_q}(n) \equiv {r_1}\left( {\bmod \,{m_1}} \right),\,\,{S_\alpha }(n) \equiv {r_2}\left( {\bmod \,{m_2}} \right)} \right\}} \right|} \cr { = {N \over {{m_1}{m_2}}} + 0\left( {{N^{1 - \delta }}} \right).} \cr } </tex-math></alternatives></disp-formula>The asymptotic relation implied by this equality was proved by Coquet, Rhin &amp; Toffin and the equality was proved for the case <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2019-0010_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mrow><mo>[</mo><mover><mn>1</mn><mo>¯</mo></mover><mo>]</mo></mrow></mrow></math><tex-math>\alpha = \left[ {\bar 1} \right]</tex-math></alternatives></inline-formula> by Spiegelhofer.</abstract>

Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.Forα=[0;1,m¯],   m≥2,\alpha = \left[ {0;\overline {1,m} } \right],\,\,\,m \ge 2,let Sα(n) denote the sum of digits in the Ostrowski α-representation of n. Let m1,m2 ≥ 2 be integers withgcd(q-1,m1)=gcd(m,m2)=1\gcd \left( {q - 1,{m_1}} \right) = \gcd \left( {m,{m_2}} \right) = 1We prove that there exists δ> 0 such that for all integers r1,r2,|{0≤n<N:Sq(n)≡r1(mod m1),  Sα(n)≡r2(mod m2)}|=Nm1m2+0(N1-δ).\matrix{ {\left| {\left\{ {0 \le n < N:{S_q}(n) \equiv {r_1}\left( {\bmod \,{m_1}} \right),\,\,{S_\alpha }(n) \equiv {r_2}\left( {\bmod \,{m_2}} \right)} \right\}} \right|} \cr { = {N \over {{m_1}{m_2}}} + 0\left( {{N^{1 - \delta }}} \right).} \cr } The asymptotic relation implied by this equality was proved by Coquet, Rhin & Toffin and the equality was proved for the case α=[1¯]\alpha = \left[ {\bar 1} \right] by Spiegelhofer.

Joint Distribution in Residue Classes of the Base-<italic>q</italic> and Ostrowski Digital Sums

Uniform distribution theory

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

Joint Distribution in Residue Classes of the Base- and Ostrowski Digital Sums

Let q be an integer greater than or equal to 2, and let Sq(n)denote the sum of digits of n in base q.Forα=[0;1,m¯],   m≥2,\alpha  = \left[ {0;\overline {1,m} }...

Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums

Published Online: Mar 27, 2020

Page range: 1 - 26

Received: Nov 16, 2017

Accepted: Dec 11, 2018

DOI: https://doi.org/10.2478/udt-2019-0010

Keywords
Ostrowski representation, Sum of digits function, Joint distribution

© 2019 Divyum Sharma, published by Sciendo

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

Joint Distribution in Residue Classes of the Base-q and Ostrowski Digital Sums

Published Online: Mar 27, 2020

Page range: 1 - 26

Received: Nov 16, 2017

Accepted: Dec 11, 2018

DOI: https://doi.org/10.2478/udt-2019-0010

KeywordsOstrowski representation, Sum of digits function, Joint distribution

© 2019 Divyum Sharma, published by Sciendo

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

Keywords
Ostrowski representation, Sum of digits function, Joint distribution