I am reading a paper on complex dynamics and Hausdorff dimension, and there is a result that I can't prove.
I have the following situation. For each $k=1,2,...$, we denote $E_k$ a finite collection of disjoint compact subsets $F$ of $\mathbb{R}^n$. We denote by $\bar{E}_k$ the compact set obtained as the union of the elements of $E_k$. We assume that every $F \in E_{k+1}$ is contained in a unique $F' \in E_k$ and that every $F \in E_k$ contains at least one element of $E_{k+1}$. Let $E= \bigcap_{k=1}^\infty \bar{E}_k$. Assume that for all $k$ and all $F \in E_k$ $ \displaystyle density(\bar{E}_{k+1}, F) = \frac{Vol(\bar{E}_{k+1} \cap F)}{Vol(F)} \geq \Delta_k$.
The problem is then to conclude that $Density(E, \bar{E}_1) \geq \prod_{k=1}^\infty \Delta_k$.
According to the author the result is easy to obtain, but don't see how to do it.
Any help will be apreciated.
I have to admit that I don't see the difficulty. According to the inequality you give: $$ Vol(\overline{E}_{k+1} \cap F) \geq \Delta_k Vol(F), $$ therefore by summing over all $F \in E_k$, $$ Vol(\overline{E}_{k+1}) = Vol(\overline{E}_{k+1} \cap \overline{E}_{k}) \geq \Delta_k Vol(\bigcup_{E_k} F) = \Delta_k Vol(\overline{E}_{k})$$ since $\overline{E}_{k+1} \subseteq \overline{E}_{k}$ and $\bigcup_{E_k} F = \overline{E}_{k}$.
This means that $$ Vol(\overline{E}_{1}) \leq \frac{1}{\Delta_1} Vol(\overline{E}_{2}) \leq \frac{1}{\Delta_1 \Delta_2} Vol(\overline{E}_{3}) \ldots$$ and the inequality for the limit should follow.