[Paper Review] On the continued fraction expansion of absolutely normal numbers

Adrian-Maria Scheerer|arXiv (Cornell University)|Jan 27, 2017

Mathematical Dynamics and Fractals11 references3 citations

TL;DR

This paper constructs an explicitly computable absolutely normal number whose continued fraction expansion is normal with respect to the Gauss-Kuzmin measure, using a recursive binary digit-by-digit construction based on Sierpiński's method and large deviations theory for mixing processes. The key contribution is the first explicit construction of a number that is simultaneously absolutely normal and continued fraction normal.

ABSTRACT

We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The construction is based on ideas of Sierpinski and uses a large deviations theorem for sums of mixing random variables.

Motivation & Objective

• To resolve the open problem of constructing an absolutely normal number that is also continued fraction normal.
• To provide a constructive, computable algorithm for generating such a number using recursive binary digit selection.
• To control measure-theoretic deviations in continued fraction normality using large deviations for mixing processes.
• To ensure the constructed number avoids non-normal sets while maintaining full measure in the desired invariant sets.
• To demonstrate that the algorithm produces a number satisfying both absolute normality and Gauss-Kuzmin normality simultaneously.

Proposed method

• Uses a recursive interval selection process on [0,1), starting with [0,1) and halving at each step to determine binary digits.
• Defines a large, full-measure set Ω of numbers with controlled partial quotient growth to ensure continued fraction normality.
• Constructs a small error set E containing non-normal numbers, with λ(E) < β for a given β > 0.
• Applies large deviations estimates for sums of mixing random variables to control deviations in the Gauss-Kuzmin measure.
• Employs finitary approximations Ω_N and E_k to make the construction computable, with error bounds ω_N and r_k.
• Selects intervals at each step where the intersection with (Ω_N \\) E_k has maximal measure, ensuring positive measure in the final set.

Experimental results

Research questions

• RQ1 Can an absolutely normal number be explicitly constructed such that its continued fraction expansion is normal with respect to the Gauss-Kuzmin measure?
• RQ2 Is it possible to design a recursive, computable algorithm that generates a number simultaneously normal in all integer bases and continued fraction normal?
• RQ3 Can large deviations theory for mixing processes be applied to control measure-theoretic deviations in continued fraction normality?
• RQ4 How can the construction be made computable while maintaining full measure in the set of desired normal numbers?
• RQ5 Does a number exist that is both absolutely normal and continued fraction normal, and can it be algorithmically generated?

Key findings

• The paper constructs a computable real number ν whose binary expansion is generated recursively by selecting intervals with maximal measure in (Ω_N \ E_k).
• The constructed number ν is absolutely normal, meaning it is simply normal to every integer base b ≥ 2.
• The continued fraction expansion of ν is normal with respect to the Gauss-Kuzmin measure, meaning all finite patterns of partial quotients appear with expected frequency.
• The construction ensures that λ((Ω \ E) ∩ c_i) > 0 at each step, guaranteeing the existence of such numbers in the selected intervals.
• The algorithm is computable in the sense that each binary digit can be determined by a finite computation using basic arithmetic and comparisons.
• The proof relies on large deviations estimates to control the measure of sets where continued fraction normality fails, ensuring the error set E has small but positive measure.