I'm trying to understand the proof of the mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step.
The Gauss map $T$ defined in the interval $I$, is given as the fractional part of $\frac{1}{x}$ where $x\in(0,1]$. If you use Kuzmin's proof of convergence to the invariant measure $\frac{1}{\ln 2}\frac{1}{1+x}dx$, you get (after deducing it for cylindrical sets)
Lemma 3: If $F \subset I$ is measurable and if $E=(x,y) \subset I$ is such that $x, y$ are $k$th convergents (associated to some $k$ continued fraction digits), then \begin{equation} |E\cap T^{-(n+k)}F| = |E||F|\left(1 + O(q^{\sqrt{n}})\right) \end{equation} where $q<1$ and the constant implicit in the big-O notation are independent of the sets and $k$ and where $|\cdot|$ denotes the invariant measure described above.
From this I want to go to
Lemma 4: If $F\subset I$ is measurable and $E=(x,y) \subset I$ is any interval, then \begin{equation} |E \cap T^{-n}F| = |E||F| + |F|O\left(q^{\sqrt{n}} \right) \end{equation} with the same constant.
Proof: Fix $k$ to be the floor of $n/2$. Find continued fraction convergents (of $k$ digits) $p^i_k, q^i_k$ such that $$ x \in \left[\frac{p^1_k}{q^1_k}, \frac{p^1_k + p^1_{k-1}}{q^1_k + p^1_{k-1}}\right] =: G_1 \text{ and } y \in \left[\frac{p^2_k}{q^2_k}, \frac{p^2_k + p^2_{k-1}}{q^2_k + p^2_{k-1}}\right] =: G_2 $$ and decompose $E$ as the disjoint union $$ E = E_1 \cup E_0 \cup E_2. $$ Here $E_1 = E \cap G_1$, $E_2 = E \cap G_2$ and $E_0$ is the interval between the endpoints of $G_1$ and $G_2$.
Now supposedly we have by lemma 3 and using that $|\cdot|$ is $T$-invariant that \begin{equation} \left||E\cap T^{-n}F| - |E||F| \right| \leq C|E_0||F|q^{\sqrt{k}} + |F|\left(|E_1| + |E_2|\right). \end{equation}
Question: How does this follow?
I think I understand where the first term comes from but not the second - \begin{equation} \begin{split} \left||E\cap T^{-n}F| - |E||F| \right| &\leq \sum^{2}_{i=0}\left| |E_i\cap T^{-n}F| - |E_i||F|\right| \\ &= C|E_0||F|q^{\sqrt{k}} + \sum^2_{i=1}||E_i\cap T^{-n}F| - |E_i||F||.\ \text{By Lemma 3.} \end{split} \end{equation} But how to estimate the second term(s)? Is it somethink trivial?
Maybe an alternate would be to apply Lemma $3$ again at this step - \begin{equation} \begin{split} ||E_1\cap T^{-n}F| - |E_1||F|| &\leq |G_1\cap T^{-n}F| + |E_1||F|\\ &\leq |G_1||F|\left(1+ Cq^{\sqrt{k}}\right) + |G_1||F| \\ &\leq |G_1||F|(1+C) \\ &\leq (1+C)|F| \frac{1}{2^{k-1}}\\ &=(1+C)|F|\left(0.5^{\frac{k-1}{\sqrt{n}}}\right)^{\sqrt{n}}\\ &=|F|O(q'^{\sqrt{n}}). \end{split} \end{equation} There doesn't seem to be any issue here, but still I cannot figure out the logic of what's written in the paper...