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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
Abstract

By a classical result of Gauss and Kuzmin, the continued fraction expansion of a “random” real number contains each digit a with asymptotic frequency log 2(1 + 1(a(a + 2))).

We generalize this result in two directions: First, for certain sets A , 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(π26).

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)kFk+2)|, where Fn 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.

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