\subsection{Gauss statistics on one typical set}
\label{sec:gauss-statistics}

Gauss cylinders connect finite itineraries with distributions of
infinite codes. Their distortion bounds give both a comparison of
measures and a proof of ergodicity. We then place all the statistical
conclusions on one Borel set.

Write $J=(0,1)$ and $J_*=J\setminus\mathbb Q$, and denote Lebesgue
measure by $\lambda$. Throughout this subsection, a word has positive integer
entries. Its length is denoted by $|\mathbf b|$, and the empty word is
allowed. We extend the Gauss map to $\R$ by
$T(x)=\{1/x\}$ for $x\ne0$ and $T(0)=0$. It is Borel, and maps $J_*$
into itself.

\subsubsection{Cylinder geometry and comparison of measures}

Fix a sequence $\mathbf a\in\mathcal D$. For its prefix of length $n$
write $G_n=g_{a_0}\circ\cdots\circ g_{a_{n-1}}$, and define
\begin{equation}
 q_{-1}=0,\qquad q_0=1,\qquad
 q_{n+1}=a_nq_n+q_{n-1}.
 \label{eq:denominator-recurrence}
\end{equation}
These denominators give both the length and the distortion of a cylinder.

\begin{lemma}
\label{lem:cylinder-density}
For $n\geq0$ and $x\geq0$,
\begin{equation}
 G_n'(x)=\frac{(-1)^n}{(q_n+q_{n-1}x)^2},\qquad
 \lambda(I_{(a_0,\ldots,a_{n-1})})
   =\frac1{q_n(q_n+q_{n-1})}.
 \label{eq:cylinder-length}
\end{equation}
Consequently, for every word $\mathbf b$ and $x\in[0,1]$,
\begin{equation}
 \frac12\lambda(I_{\mathbf b})\leq |G_{\mathbf b}'(x)|
 \leq2\lambda(I_{\mathbf b}).
 \label{eq:cylinder-density}
\end{equation}
Moreover, for $x,y\geq0$,
\begin{equation}
 \left|\log|G_{\mathbf b}'(x)|-\log|G_{\mathbf b}'(y)|\right|
 \leq2|x-y|.
 \label{eq:log-prefix-lipschitz}
\end{equation}
\end{lemma}

\begin{proof}
Induction in~\eqref{eq:denominator-recurrence} gives
$q_n\geq1$ and $0\leq q_{n-1}\leq q_n$. Put
$D_n(x)=q_n+q_{n-1}x$. The recurrence is equivalent to
\[
 D_{n+1}(x)=(a_n+x)D_n(g_{a_n}(x)).
\]
Since $G_{n+1}=G_n\circ g_{a_n}$, the derivative formula follows by
the chain rule, starting from $G_0(x)=x$.
Each branch is injective, and hence so is $G_n$. Integrating its absolute
derivative on $J$ gives the cylinder length. In fact,
\[
 \frac{d}{dx}\frac{x}{q_n(q_n+q_{n-1}x)}
   =\frac1{(q_n+q_{n-1}x)^2}.
\]
For $x\in[0,1]$, the derivative divided by the cylinder length lies
between $q_n/(q_n+q_{n-1})$ and $(q_n+q_{n-1})/q_n$. These bounds
imply~\eqref{eq:cylinder-density}. Finally,
\[
 \frac{d}{dx}\log|G_n'(x)|
   =-\frac{2q_{n-1}}{q_n+q_{n-1}x}
\]
has absolute value at most $2$ on $[0,\infty)$. The mean value theorem
proves~\eqref{eq:log-prefix-lipschitz}.
\end{proof}

Every $z\in J_*$ has a code. Indeed, all its iterates remain in $J_*$.
Taking $a_m=\lfloor1/T^mz\rfloor$ gives
$T^mz=g_{a_m}(T^{m+1}z)$. Proposition~\ref{prop:phase-comparison}, with
zero errors, yields $z=c(\mathbf a)$.

Two intersecting cylinders have comparable words: one word is a prefix
of the other. To see this, for $x\in J$ the first digit of $g_k(x)$ is
$\lfloor1/g_k(x)\rfloor=k$. Thus intersecting nonempty words have the
same first digit. Cancel its inverse branch and repeat. In particular,
the cylinders of a fixed length are pairwise disjoint, and their union
contains $J_*$. They have total Lebesgue measure one. Also,
\begin{equation}
 c(\mathbf a)\in I_{\mathbf b}
 \quad\Longleftrightarrow\quad
 (a_0,\ldots,a_{|\mathbf b|-1})=\mathbf b.
 \label{eq:code-cylinder}
\end{equation}

\begin{lemma}[Cylinder domination]
\label{lem:cylinder-domination}
Let $\mu$ and $\xi$ be Borel measures on $\R$, with $\xi$ outer regular.
Suppose that $\mu(\R\setminus J_*)=0$ and
\[
 \mu(I_{\mathbf b})\leq\xi(I_{\mathbf b})
 \qquad\text{for every nonempty word }\mathbf b.
\]
Then $\mu\leq\xi$.
\end{lemma}

\begin{proof}
Let $O\subset\R$ be open. Take all nonempty words $\mathbf b$ such that
$I_{\mathbf b}\subset O$, but no nonempty proper prefix has this
property. Their cylinders are pairwise disjoint by comparability.
They cover $O\cap J_*$: for $z=c(\mathbf a)\in O$, the cylinders of its
successive prefixes contain $z$ and have diameter at most
$4^{-\lfloor n/2\rfloor}$. Some cylinder is therefore contained in $O$,
and there is a first nonempty prefix with this property.
Writing $\mathcal W(O)$ for these words, countable subadditivity gives
\[
 \mu(O)\leq\sum_{\mathbf b\in\mathcal W(O)}\mu(I_{\mathbf b})
 \leq\sum_{\mathbf b\in\mathcal W(O)}\xi(I_{\mathbf b})
 =\xi\!\left(\bigcup_{\mathbf b\in\mathcal W(O)}I_{\mathbf b}\right)
 \leq\xi(O).
\]
For a Borel set $A$, take the infimum over open supersets of $A$ and use
outer regularity of $\xi$.
\end{proof}

The useful feature of this argument is that the covering cylinders can
have different lengths. A bound at every finite depth consequently
controls arbitrary Borel sets without a limiting boundary estimate.

\begin{corollary}[Distribution of codes]
\label{cor:code-domination}
Let $(X,\mu)$ be a measure space and let
$\mathbf a:X\to\mathcal D$ be measurable. If $C\geq0$ and
\begin{equation}
 \mu\{q:(a_0(q),\ldots,a_{r-1}(q))=\mathbf b\}
 \leq C\lambda(I_{\mathbf b})
 \label{eq:prefix-mass-bound}
\end{equation}
for every nonempty word $\mathbf b$ of length $r$, then
\begin{equation}
 (c\circ\mathbf a)_*\mu\leq C\lambda|_J.
 \label{eq:code-domination}
\end{equation}
In particular the distribution of codes is absolutely continuous.
\end{corollary}

\begin{proof}
The map $c\circ\mathbf a$ is measurable and takes its values in $J_*$.
By~\eqref{eq:code-cylinder}, its pushforward satisfies the cylinder
bounds of Lemma~\ref{lem:cylinder-domination}, with
$\xi=C\lambda|_J$.
\end{proof}

\subsubsection{Ergodicity and a common typical set}

The density in~\eqref{eq:gauss-measure} is strictly positive and satisfies
\begin{equation}
 \frac1{2\log2}\leq\frac1{\log2\,(1+x)}\leq\frac1{\log2}
 \qquad(0\leq x\leq1).
 \label{eq:gauss-density-bounds}
\end{equation}
Its primitive is $\log(1+x)/\log2$. Thus $\nu$ is a probability,
has the same null sets as $\lambda|_J$, and gives no mass to rational
points. For $0\leq a\leq b\leq1$,
\begin{equation}
 \nu((a,b))=\frac{\log(1+b)-\log(1+a)}{\log2}.
 \label{eq:gauss-interval}
\end{equation}

\begin{proposition}
\label{prop:gauss-ergodic}
The Gauss map preserves $\nu$ and is ergodic with respect to $\nu$.
\end{proposition}

\begin{proof}
Write $w(x)=1/(\log2\,(1+x))$. For $k\geq1$ and $x\in J$,
\[
 w(g_k(x))|g_k'(x)|
 =\frac1{\log2}\left(\frac1{k+x}-\frac1{k+1+x}\right).
\]
Summing over $k$ telescopes to $w(x)$. For a Borel set $A\subset J$,
the sets $g_k(A)$ are disjoint and their union agrees with $T^{-1}A$
on $J$ up to rational points. Change of variables in each branch and
monotone convergence therefore give
\[
 \nu(T^{-1}A)=\sum_{k\geq1}\nu(g_k(A))
 =\int_A\sum_{k\geq1}w(g_k(x))|g_k'(x)|\dd x=\nu(A).
\]
Since $\nu$ and its image under $T$ are concentrated on $J$, this proves
invariance for every Borel set in $\R$.

For ergodicity, let $E\subset\R$ be Borel with $T^{-1}E=E$, and put
$B=E\cap J$. Invariance of membership, applied successively to inverse
branches, gives
$G_{\mathbf b}(B)=I_{\mathbf b}\cap E$.
Lemma~\ref{lem:cylinder-density} and change of variables imply
\[
 \lambda(I_{\mathbf b}\cap E)
   \geq\tfrac12\lambda(I_{\mathbf b})\lambda(B).
\]
Apply Lemma~\ref{lem:cylinder-domination} to
$\mu=\tfrac12\lambda(B)\lambda|_J$ and $\xi=\lambda|_B$.
The latter is a finite Borel measure on $\R$, hence outer regular.
It follows that
\[
 \tfrac12\lambda(B)\lambda|_J\leq\lambda|_B.
\]
Evaluating this inequality on $E^c$ yields
$\lambda(B)\lambda(J\setminus E)=0$. Thus $E$ is null or conull in
$J$ for Lebesgue measure, and hence for $\nu$ by
\eqref{eq:gauss-density-bounds}. This is ergodicity.
\end{proof}

We use Birkhoff's pointwise ergodic theorem in the following form.
For an ergodic measure-preserving map $F$ of a probability space
$(X,\mu)$ and a real integrable function $f$,
\begin{equation}
 \frac1N\sum_{m=0}^{N-1}f(F^mq)\longrightarrow\int_X f\dd\mu
 \quad\text{for }\mu\text{-almost every }q.
 \label{eq:birkhoff-input}
\end{equation}
See~\cite[Corollary~10.1.2, p.~89]{Walkden}.
Here it is applied to the countable family of cylinder indicators.

Let $\mathcal B$ consist of the points $z\in\R$ such that, for every
word $\mathbf b$,
\begin{equation}
 \frac1N\sum_{m=0}^{N-1}\ind_{I_{\mathbf b}}(T^mz)
 \longrightarrow\nu(I_{\mathbf b}).
 \label{eq:cylinder-typical}
\end{equation}
Each convergence condition is Borel. Proposition~\ref{prop:gauss-ergodic}
and~\eqref{eq:birkhoff-input} show that $\nu(\mathcal B)=1$, since the
indicators are bounded and there are only countably many words.
Moreover, $T^{-1}\mathcal B=\mathcal B$: shifting an average of an
indicator changes it by at most $1/N$.

We next pass from cylinder indicators to all continuous functions on
$[0,1]$ on this same set. The following elementary observation controls
the infinitely many cylinders at a fixed depth. If probability masses
$p_N(i)$ on a countable set converge pointwise to probability masses
$p(i)$, then
\begin{equation}
 \sum_i|p_N(i)-p(i)|
   =2-2\sum_i\min(p_N(i),p(i))\longrightarrow0.
 \label{eq:discrete-l1}
\end{equation}
Indeed, the summands in the last sum converge to $p(i)$ and are bounded
by the summable sequence $p(i)$. Dominated convergence proves the limit.

\begin{proposition}
\label{prop:typical-statistics}
If $\mathbf a\in\mathcal D$ and $z=c(\mathbf a)\in\mathcal B$, then
all six limits~\eqref{eq:phase-law}--\eqref{eq:infinite-mean} hold with
$K_m=a_m$ and $x_m=T^mz$.
\end{proposition}

\begin{proof}
The block frequencies follow from~\eqref{eq:code-cylinder}, the shift
identity~\eqref{eq:code-recurrence}, and the definition of $\mathcal B$.
For continuous observations, fix a depth $r$ and write
\[
 p_N(\mathbf b)=\frac1N\#\{m<N:T^mz\in I_{\mathbf b}\},\qquad
 p(\mathbf b)=\nu(I_{\mathbf b}),\qquad |\mathbf b|=r.
\]
The depth $r$ cylinders partition $J_*$, so these are probability masses.
They converge pointwise by~\eqref{eq:cylinder-typical}, hence in
$\ell^1$ by~\eqref{eq:discrete-l1}.

For each cylinder take the representative
$\xi_{\mathbf b}=G_{\mathbf b}(0)\in[0,1]$.
Every point of that cylinder is within $4^{-\lfloor r/2\rfloor}$ of
its representative. If $f\in C([0,1])$, choose $r$ sufficiently large
that the oscillation of $f$ over this distance is at most $\varepsilon$.
Then
\begin{equation}
 \left|\frac1N\sum_{m<N}f(T^mz)-\int f\dd\nu\right|
 \leq2\varepsilon+
 \|f\|_\infty\sum_{|\mathbf b|=r}|p_N(\mathbf b)-p(\mathbf b)|.
 \label{eq:continuous-cylinder-approximation}
\end{equation}
The two approximation errors come from replacing $f$ by its representative
value in the empirical average and in the integral. Let $N\to\infty$,
then $\varepsilon\to0$. This proves the continuous observation law.

For one digit, $I_{(k)}=(1/(k+1),1/k)$.
Formula~\eqref{eq:gauss-interval} gives
\[
 \nu(I_{(k)})=\frac1{\log2}\log\frac{(k+1)^2}{k(k+2)}.
\]
At a fixed threshold $k$ the tail indicator is one minus the sum of the
first $k-1$ digit indicators. Its frequency is therefore
\[
 1-\sum_{j=1}^{k-1}\nu(I_{(j)})
 =\nu((0,1/k))=\frac{\log(1+1/k)}{\log2}.
\]
Finally, the identity
$\min(a_m,R)=\sum_{k=1}^R\ind_{\{a_m\geq k\}}$ gives the truncated
mean. The limiting sum telescopes:
\[
 \sum_{k=1}^R\frac{\log(1+1/k)}{\log2}
   =\frac{\log(R+1)}{\log2}.
\]
The average of $a_m$ is bounded below by every truncated average.
Given any real bound, first take $R$ for which the last expression exceeds
that bound, and then take $N$ large. This proves divergence of the full
mean to $+\infty$.
\end{proof}

Finite-cylinder approximation applies to every continuous observation
on the same set $\mathcal B$. A sequence with code in $\mathcal B$ cannot have
bounded digits: any omitted digit has a strictly positive prescribed
frequency. Equivalence of $\nu$ and $\lambda|_J$ thus shows that the
set of codes with bounded digits is Lebesgue null.

\begin{corollary}[Statistical transfer]
\label{cor:statistical-transfer}
Let $(X,\mu)$ be a measure space, let $\mathbf a:X\to\mathcal D$ be
measurable, and let $y_m(q)\in[0,1]$ for $\mu$-almost every $q$ and all
$m$. Suppose that
\[
 (c\circ\mathbf a)_*\mu\ll\lambda|_J,\qquad
 |y_m(q)-g_{a_m(q)}(y_{m+1}(q))|\longrightarrow0
 \quad\text{for }\mu\text{-almost every }q.
\]
Then, on one set of full $\mu$ measure, all six
limits~\eqref{eq:phase-law}--\eqref{eq:infinite-mean} hold with
$K_m=a_m(q)$ and $x_m=y_m(q)$.
\end{corollary}

\begin{proof}
Absolute continuity puts the code in $\mathcal B$ for almost every $q$.
Intersect with the full measure set on which the phase conditions hold.
Proposition~\ref{prop:typical-statistics} gives every digit statistic;
Corollary~\ref{cor:continuous-transfer} gives the continuous observation
law for the phases $y_m(q)$.
\end{proof}