In my notes I have the following claim:
Let $Q \subseteq \mathbb{R}^n$ be a $d$-dimensional set and $L_j$ be a $j$-dimensional plane in $\mathbb{R}^n$ $(j<d)$ such that $$ \text{dist}(y,L_j) \leq A^{-1}\text{diam}(Q) \qquad \text{for every } y \in Q, $$ Then we can cover $Q$ with no more than $(A+1)^{j}$ many balls of radius $\frac{10n}{A} \text{diam Q}$.
Here $d$ is the Hausdorff dimension. Or even more - quantitatively: There is a constant $C$ (independent of the set) such that the $d$-dimensional Hausdorff measure of $Q\cap B(x,R)\sim R^d$ for any $x\in Q,R>0$.
Could someone explain please why is it true?