<abstract xmlns="http://www.w3.org/1999/xhtml"> Denote by Θ1,Θ2, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].</abstract>

Denote by Θ1,Θ2, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For almost all x this is a uniformly distributed sequence in the interval [0, 1/2 ]. The average distance between two consecutive terms of this sequence and their correlation coefficient are explicitly calculated and it is shown why these two values are close to 1/6 and 0, respectively, the corresponding values for a random sequence in [0, 1/2].

A Note on the Continued Fraction of Minkowski

Uniform distribution theory

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

{"article-title":"A Note on the Continued Fraction of Minkowski"}

Denote by Θ1,Θ2, · · · the sequence of approximation coefficients of Minkowski’s diagonal continued fraction expansion of a real irrational number x. For...

A Note on the Continued Fraction of Minkowski

Published Online: Jan 30, 2018

Page range: 125 - 130

Received: Oct 10, 2016

Accepted: Feb 15, 2017

DOI: https://doi.org/10.1515/udt-2017-0019

Keywords
Continued fractions, approximation coefficients, metrical theory

© 2018

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

A Note on the Continued Fraction of Minkowski

Published Online: Jan 30, 2018

Page range: 125 - 130

Received: Oct 10, 2016

Accepted: Feb 15, 2017

DOI: https://doi.org/10.1515/udt-2017-0019

KeywordsContinued fractions, approximation coefficients, metrical theory

© 2018

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

Keywords
Continued fractions, approximation coefficients, metrical theory