An Elementary Characterization of the Gauss--Kuzmin Measure in the Theory of Continued Fractions

By a classical result of Gauss and Kuzmin, the frequency with which a string $\mathbf{a}=(a_1,\dots ,a_n)$ of positive integers appears in the continued fraction expansion of a random real number is given by ${\mu }_{GK}(I(\mathbf{a}))$, where $I(\mathbf{a})$ is the set of real numbers in $[0,1)$ whose continued fraction expansion begins with the string $\mathbf{a}$ and ${\mu }_{GK}$ is the \emph{Gauss–Kuzmin measure}, defined by ${\mu }_{GK}(I)=\frac{1}{\log 2}{\int }_I\frac{1}{1+x}dx$, for any interval $I\subseteq [0,1]$. % It is known that the Gauss–Kuzmin measure satisfies the symmetry property $(\ast )$ ${\mu }_{GK}(I(\mathbf{a}))={\mu }_{GK}(I(\overleftarrow{\mathbf{a}}))$, where $\overleftarrow{\mathbf{a}}=(a_n,\dots ,a_1)$ is the reverse of the string $\mathbf{a}$. We show that this property in fact characterizes the Gauss–Kuzmin measure: If $\mu$ is any probability measure with continuous density function on $[0,1]$ satisfying $\mu (I(\mathbf{a}))=\mu (I(\overleftarrow{\mathbf{a}}))$ for all finite strings $\mathbf{a}$, then $\mu ={\mu }_{GK}$. % We also consider the question whether symmetries analogous to $(\ast )$ hold for permutations of $\mathbf{a}$ other than the reverse $\overleftarrow{\mathbf{a}}$; we call such a symmetry \emph{nontrivial}. We show that strings $\mathbf{a}$ of length $3$ have no nontrivial symmetries, while for each $n\ge 4$ there exists an infinite family of strings $\mathbf{a}$ of length $n$ that do have nontrivial symmetries. Finally we present numerical data supporting the conjecture that, in an appropriate asymptotic sense, ``almost all’’ strings $\mathbf{a}$ have no nontrivial symmetries.