An Elementary Characterization of the Gauss--Kuzmin Measure in the Theory of Continued Fractions
Shreyas Singh, Zhuo Zhang, AJ Hildebrand
TL;DR
The paper provides an elementary characterization of the Gauss--Kuzmin measure $μ_{GK}$ by showing that any probability measure on $[0,1]$ with continuous density that equates the frequencies of a CF block with those of its reverse must be $μ_{GK}$. It develops and leverages convergent matrices and fundamental intervals to prove this reversal-symmetry criterion, and then establishes optimality results showing the condition cannot be weakened to finitely many lengths. It further analyzes nontrivial symmetries, proving there are no length-3 counterexamples but constructing infinite families for lengths $n\ge4$, supported by numerical data suggesting such exceptional strings are rare. The work advances understanding of how CF digit order affects block frequencies and offers concrete questions about the prevalence of nontrivial symmetries and their asymptotics. The results blend elementary CF theory with measure-theoretic reasoning, yielding a clear criterion for GK measure and insight into the symmetry structure of CF expansions.$
Abstract
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 $μ_{GK}({I(\mathbf{a})})$, where $I(\mathbf{a})$ is the set of real numbers in $[0,1)$ whose continued fraction expansion begins with the string $\mathbf{a}$ and $μ_{GK}$ is the \emph{Gauss--Kuzmin measure}, defined by $μ_{GK}(I)= \frac{1}{\log 2}\int_I \frac{1}{1+x} dx$, for any interval $I\subseteq[0,1]$. % It is known that the Gauss--Kuzmin measure satisfies the symmetry property $(*)$ $μ_{GK}(I(\mathbf{a}))=μ_{GK}(I(\overleftarrow{\mathbf{a}}))$, 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 $μ$ is any probability measure with continuous density function on $[0,1]$ satisfying $μ(I(\mathbf{a}))=μ(I(\overleftarrow{\mathbf{a}}))$ for all finite strings $\mathbf{a}$, then $μ=μ_{GK}$. % We also consider the question whether symmetries analogous to $(*)$ 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.