On the Continued Fraction Expansion of Almost All Real Numbers

• Published in: arXiv.org 2024-03
• Main Authors: Jin, Alex, Singh, Shreyas, Zhang, Zhuo, Hildebrand, A J
• Format: Article
• Language: English
• Subjects: Digits; Fibonacci numbers; Real numbers
• ISSN: 2331-8422

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 \(\log_2(\pi^2/6)\). 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 \(\pi\), and we provide strong statistical evidence that the continued fraction expansion of \(\pi\) behaves like that of a random real number.