<abstract xmlns="http://www.w3.org/1999/xhtml">

The Minkowski question-mark function ?(<italic>x</italic>) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this function, if exists, can take only two values: 0 and +∞.It isalso known that the value of the derivative ? (<italic>x</italic>)atthe point <italic>x</italic> =[0; <italic>a</italic>1,<italic>a</italic>2,...,<italic>a</italic><italic>t</italic>,...] is connected with the limit behaviour of the arithmetic mean (<italic>a</italic>1 +<italic>a</italic>2 +···+<italic>a</italic><italic>t</italic>)<italic>/t</italic>. Particularly, N. Moshchevitin and A. Dushistova showed that if
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0014_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><msub><mrow><mi>a</mi></mrow><mn>1</mn></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mn>2</mn></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mi>t</mi></msub><mo>&lt;</mo><msub><mrow><mi>κ</mi></mrow><mn>1</mn></msub><mo>,</mo></mrow></math>
<tex-math>{a_1} + {a_2} + \cdots + {a_t} &lt; {\kappa _1},</tex-math>
</alternatives>
</disp-formula>
where <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2022-0014_eq_002.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>κ</mi></mrow><mn>1</mn></msub><mo>=</mo><mn>2</mn><mo>log</mo><mrow><mo>(</mo><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mn>5</mn></msqrt></mrow><mn>2</mn></mfrac></mrow><mo>)</mo></mrow><mo>/</mo><mo>log</mo><mn>2</mn><mo>=</mo><mn>1.3884</mn><mo>…</mo></mrow></math>
<tex-math>{\kappa _1} = 2\log \left( {{{1 + \sqrt 5 } \over 2}} \right)/\log 2 = 1.3884 \ldots</tex-math>
</alternatives>
</inline-formula>, then ?′(<italic>x</italic>)=+∞.They also proved that the constant <italic>κ</italic>1 is non-improvable. We consider a dual problem: how small can be the quantity <italic>a</italic>1 + <italic>a</italic>2 + ··· + <italic>a</italic><italic>t</italic> − <italic>κ</italic>1<italic>t</italic> if we know that ? (<italic>x</italic>) = 0? We obtain the non-improvable estimates of this quantity.
</abstract>

The Minkowski question-mark function ?(x) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this function, if exists, can take only two values: 0 and +∞.It isalso known that the value of the derivative ? (x)atthe point x =[0; a1,a2,...,at,...] is connected with the limit behaviour of the arithmetic mean (a1 +a2 +···+at)/t. Particularly, N. Moshchevitin and A. Dushistova showed that if



a1+a2+⋯+at<κ1,
{a_1} + {a_2} + \cdots + {a_t} < {\kappa _1},


where 


κ1=2log(1+52)/log2=1.3884…
{\kappa _1} = 2\log \left( {{{1 + \sqrt 5 } \over 2}} \right)/\log 2 = 1.3884 \ldots

, then ?′(x)=+∞.They also proved that the constant κ1 is non-improvable. We consider a dual problem: how small can be the quantity a1 + a2 + ··· + at − κ1t if we know that ? (x) = 0? We obtain the non-improvable estimates of this quantity.

On the Derivative of the Minkowski Question-Mark Function

National Research University Higher School of Economics

Uniform distribution theory

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

{"article-title":"On the Derivative of the Minkowski Question-Mark Function"}

The Minkowski question-mark function ?(x) is a continuous monotonous function defined on [0, 1] interval. It is well known fact that the derivative of this...

On the Derivative of the Minkowski Question-Mark Function

Publié en ligne: 12 déc. 2022

Pages: 101 - 126

Reçu: 28 juil. 2022

Accepté: 20 août 2022

DOI: https://doi.org/10.2478/udt-2022-0014

Mots clés
Minkowski question-mark function, Continued fractions

© 2022 Dmitry Gayfulin, published by Sciendo

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

On the Derivative of the Minkowski Question-Mark Function

Publié en ligne: 12 déc. 2022

Pages: 101 - 126

Reçu: 28 juil. 2022

Accepté: 20 août 2022

DOI: https://doi.org/10.2478/udt-2022-0014

Mots clésMinkowski question-mark function, Continued fractions

© 2022 Dmitry Gayfulin, published by Sciendo

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

Mots clés
Minkowski question-mark function, Continued fractions