# (Exponential) Mixing property for Gauss map - going from cylinders to intervals

I'm trying to understand the proof of a 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 $$$$|E\cap T^{-(n+k)}F| = |E||F|\left(1 + O(q^{\sqrt{n}})\right)$$$$ where $$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 $$$$|E \cap T^{-n}F| = |E||F| + |F|O\left(q^{\sqrt{n}} \right)$$$$ 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 $$$$\left||E\cap T^{-n}F| - |E||F| \right| \leq C|E_0||F|q^{\sqrt{k}} + |F|\left(|E_1| + |E_2|\right).$$$$

Question: How does this follow?

I think I understand where the first term comes from but not the second - $$$$\begin{split} \left||E\cap T^{-n}F| - |E||F| \right| &\leq \sum^{2}_{i=0}\left| |E_i\cap T^{-n}F| - |E_i||F|\right| \\ &= C|E_0||F|q^{\sqrt{k}} + \sum^2_{i=1}||E_i\cap T^{-n}F| - |E_i||F||.\ \text{By Lemma 3.} \end{split}$$$$ But how to estimate the second term(s)? Is it somethink trivial?

The factor $$|F|$$ in Lemma $$4$$ seems to be crucial when applying it elsewhere in the paper.

Maybe an alternate would be to apply Lemma $$3$$ again at this step - $$$$\begin{split} ||E_1\cap T^{-n}F| - |E_1||F|| &\leq |G_1\cap T^{-n}F| + |E_1||F|\\ &\leq |G_1||F|\left(1+ Cq^{\sqrt{k}}\right) + |G_1||F| \\ &\leq |G_1||F|(1+C) \\ &\leq (1+C)|F| \frac{1}{2^{k-1}}\\ &=(1+C)|F|\left(0.5^{\frac{k-1}{\sqrt{n}}}\right)^{\sqrt{n}}\\ &=|F|O(q'^{\sqrt{n}}). \end{split}$$$$ There doesn't seem to be any issue here, but still I cannot figure out the logic of what's written in the paper...