On the Continued Fraction Expansion of Almost All Real Numbers

TL;DR

This work advances the metric theory of continued fractions by providing explicit closed-form frequencies for CF digits drawn from special sets and strings, extending Gauss-Kuzmin theory. It proves that the frequency of digits of the form $p^k-1$ equals $\log_2\zeta(2k)$ and that digits of the form $n^2-1$ have frequency $\log_2\left(\frac{8\pi}{e^{\pi}-e^{-\pi}}\right)$, while run frequencies for identical-digit strings are given by a simple two-term recurrence rooted in Fibonacci-like sequences. A key result expresses run frequencies as $P(\underbrace{(a,\dots,a)}_k)=\left|\log_2\left(1+\frac{(-1)^k}{\left(F^{(a)}_{k+2}\right)^2}\right)\right|$, leading to precise asymptotics for runs of 1s via Fibonacci numbers. The paper further validates these theoretical predictions by analyzing 300 million digits of $\\pi$, showing no statistically significant deviations from random-CF behavior and providing strong empirical support for the conjectured randomness of $\\pi$’s continued fraction expansion.

Abstract

By a classical result of Gauss and Kuzmin, the continued fraction expansion of a ``random'' real number contains each digit $a\in\mathbb{N}$ with asymptotic frequency $\log_2(1+1/(a(a+2)))$. We generalize this result in two directions: First, for certain sets $A\subset\mathbb{N}$, we establish simple explicit formulas for the frequency with which the continued fraction expansion of a random real number contains a digit from the set $A$. For example, we show that digits of the form $p-1$, where $p$ is prime, appear with frequency $\log_2(π^2/6)$. Second, we obtain a simple formula for the frequency with which a string of $k$ consecutive digits $a$ appears in the continued fraction expansion of a random real number. In particular, when $a=1$, this frequency is given by $|\log_2(1+(-1)^k/F_{k+2})|$, where $F_n$ is the $n$th Fibonacci number. Finally, we compare the frequencies predicted by these results with actual frequencies found among the first 300 million continued fraction digits of $π$, and we provide strong statistical evidence that the continued fraction expansion of $π$ behaves like that of a random real number.

Figures (2)

• Figure 1: The Gauss-Kuzmin distribution $P(a)=\log_2(1+\frac{1}{a(a+2)})$, $a=1,2,\dots$. ($\log_2 t =\log t/\log 2$ denotes the base $2$ logarithm.)
• Figure 2: Distribution of z-scores for digits of the form $p-1$ (top row) and $n^2-1$ (bottom row), corresponding to block sizes 250,000 (left figure), 500,000 (middle figure), and 1,000,000 (right figure),

Theorems & Definitions (13)

• Theorem 1: Shifted prime powers
• Theorem 2: Shifted squares
• Theorem 3: Strings of identical digits
• Corollary 4
• Lemma 2.1: Gauss-Kuzmin Theorem, iosifescu-kraaikamp-book
• Lemma 2.2: Generalized Gauss-Kuzmin Theorem, iosifescu-kraaikamp-book
• Proposition 3.1: Explicit formula for subset frequencies
• proof
• proof : Proof of Theorem \ref{['thm:Pk']}
• proof : Proof of Theorem \ref{['thm:S']}