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

TL;DR

This paper characterizes the Gauss--Kuzmin measure uniquely by its symmetry property involving reversed continued fraction strings and explores the existence of other symmetries for strings of various lengths, supported by numerical evidence.

Contribution

It proves that the Gauss--Kuzmin measure is uniquely determined by its symmetry property and investigates the presence of nontrivial symmetries for strings of different lengths.

Findings

Gauss--Kuzmin measure is uniquely characterized by its symmetry property.

No nontrivial symmetries for strings of length 3.

Existence of infinite families of strings with nontrivial symmetries for lengths ≥4.

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 $\mu_{GK}({I(\mathbf{a})})$, 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 $(*)$ $\mu_{GK}(I(\mathbf{a}))=\mu_{GK}(I(\overleftarrow{\mathbf{a}}))$, 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 $(*)$ 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.