On the continued fraction expansion of almost all real numbers

Jin, Alex; Singh, Shreyas; Zhang, Zhuo; Hildebrand, A.J.

doi:10.2140/involve.2025.18.583

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On the continued fraction expansion of almost all real numbers

Alex Jin, Shreyas Singh, Zhuo Zhang and A.J. Hildebrand

Vol. 18 (2025), No. 4, 583–600

DOI: 10.2140/involve.2025.18.583

Abstract

By a classical result of Gauss and Kuzmin, the continued fraction expansion of a “random” real number contains each digit $a \in ℕ$ with asymptotic frequency $\log_{2} (1 + 1 ∕ (a (a + 2)))$ .

We generalize this result in two directions: First, for certain sets $A \subset ℕ$ , 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.

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Keywords

continued fraction, digits, random number, $\pi$

Mathematical Subject Classification

Primary: 11K50, 11A55

Milestones

Received: 21 December 2022

Revised: 26 February 2024

Accepted: 12 March 2024

Published: 30 July 2025

Communicated by Filip Saidak

Authors

Alex Jin
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

Shreyas Singh
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

Zhuo Zhang
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

A.J. Hildebrand
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

Vol. 18, No. 4, 2025