<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>α</italic> be an irrational real number; the behaviour of the sum <italic>SN</italic> (<italic>α</italic>):= (<italic>−</italic>1)[<italic>α</italic>] +(<italic>−</italic>1)[2<italic>α</italic>] + <italic>···</italic> +(<italic>−</italic>1)[<italic>Nα</italic>] depends on the continued fraction expansion of <italic>α/</italic>2. Since the continued fraction expansion of <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msqrt><mn>2</mn></msqrt><mo>/</mo><mn>2</mn></mrow></math><tex-math>\sqrt 2 /2</tex-math></alternatives></inline-formula> has bounded partial quotients, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>N</mi></msub><mrow><mo>(</mo><mrow><msqrt><mn>2</mn></msqrt></mrow><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mo>log</mo><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math><tex-math>{S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right)</tex-math></alternatives></inline-formula> and this bound is best possible. The partial quotients of the continued fraction expansion of <italic>e</italic> grow slowly and thus <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>N</mi></msub><mrow><mo>(</mo><mrow><mn>2</mn><mi>e</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo>log</mo><msup><mrow><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow><mrow><mo>log</mo><mi> </mi><mo>log</mo><msup><mrow><mrow><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math><tex-math>{S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right)</tex-math></alternatives></inline-formula>, again best possible. The partial quotients of the continued fraction expansion of <italic>e/</italic>2 behave similarly as those of <italic>e</italic>. Surprisingly enough <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_udt-2020-0007_eq_004.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>S</mi></mrow><mi>N</mi></msub><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>=</mo><mi>O</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo>log</mo><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow><mrow><mo>log</mo><mi> </mi><mo>log</mo><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>)</mo></mrow></mrow></math><tex-math>1188</tex-math></alternatives></inline-formula>.</abstract>

Let α be an irrational real number; the behaviour of the sum SN (α):= (−1)[α] +(−1)[2α] + ··· +(−1)[Nα] depends on the continued fraction expansion of α/2. Since the continued fraction expansion of 2/2\sqrt 2 /2 has bounded partial quotients, SN(2)=O(log(N)){S_N}\left( {\sqrt 2 } \right) = O\left( {\log \left( N \right)} \right) and this bound is best possible. The partial quotients of the continued fraction expansion of e grow slowly and thus SN(2e)=O(log(N)2log log(N)2){S_N}\left( {2e} \right) = O\left( {{{\log {{\left( N \right)}^2}} \over {\log \,\log {{\left( N \right)}^2}}}} \right), again best possible. The partial quotients of the continued fraction expansion of e/2 behave similarly as those of e. Surprisingly enough SN(e)=O(log(N)log log(N))1188.

A curiosity About (<italic>−</italic>1)[<italic>e</italic>] +(<italic>−</italic>1)[2<italic>e</italic>] + <italic>···</italic> +(<italic>−</italic>1)[<italic>Ne</italic>]

Uniform distribution theory

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

Let α be an irrational real number; the behaviour of the sum SN (α):= (−1)[α] +(−1)[2α] + ··· +(−1)[Nα] depends on the continued fraction expansion of α/2....

A curiosity About (−1)^[^e^] +(−1)^[2^e^] + ··· +(−1)^[^Ne^]

Online veröffentlicht: 25. Dez. 2020

Seitenbereich: 1 - 8

Eingereicht: 05. Juni 2020

Akzeptiert: 17. Juni 2020

DOI: https://doi.org/10.2478/udt-2020-0007

Schlüsselwörter
Oscillating sums, uniform distribution modulo 1

© 2020 Francesco Amoroso et al., published by Sciendo

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

A curiosity About (−1)[e] +(−1)[2e] + ··· +(−1)[Ne]

Online veröffentlicht: 25. Dez. 2020

Seitenbereich: 1 - 8

Eingereicht: 05. Juni 2020

Akzeptiert: 17. Juni 2020

DOI: https://doi.org/10.2478/udt-2020-0007

SchlüsselwörterOscillating sums, uniform distribution modulo 1

© 2020 Francesco Amoroso et al., published by Sciendo

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

A curiosity About (−1)^[^e^] +(−1)^[2^e^] + ··· +(−1)^[^Ne^]

Schlüsselwörter
Oscillating sums, uniform distribution modulo 1