On the Continued Fraction Expansion of Almost All Real Numbers

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 $\log_2(\pi^2/6)$. 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.