On convergence rate in the Gauss–Kuzmin problem for θ-expansions

Continued fractions arise naturally in long division and the theory of the approximation to real numbers by rational numbers. This research considered the implementation on the convergent of θ-expansions of real numbers of x∈(0,θ) with 0<θ<1. The convergent of θ-expansions are also called as θ-convergent of continued fraction expansions. This study aimed to establish the properties for a family of θ-convergent in θ-expansions. The idea of discovering the behaviours of θ-convergent evolved from the concept of regular continued fraction (RCF) expansion and sequences involved in θ-expansions. The θ-expansions algorithm was used to compute the values of θ-convergent with the help of Maple software. Consequently, it proved to be an efficient method for fast computer implementation. The growth rate of θ-convergent was investigated to highlight the performance of θ-convergent. The analysis on θ-convergent revealed the convergent that gives a better approximation yielding to fewer convergence errors. This whole paper thoroughly derived the behaviours of θ-convergent, which measure how well a number x is approximated by its convergents for almost all irrational numbers.

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On convergence rate in the Gauss–Kuzmin problem for θ-expansions

Cited by 3 publications

References 12 publications

A Two-Dimensional Gauss–Kuzmin Theorem Associated with the Random Fibonacci-Type Sequences

A Two-Dimensional Gauss–Kuzmin Theorem Associated with the Random Fibonacci-Type Sequences

The Hausdorff Dimension of the Sets of Irrationals with Prescribed Relative Growth Rates

The behaviours of convergents in theta-expansions: computational insights based on theta-expansions algorithm using the Maple software

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