I am studying fractals and have come to a section about the intersection of fractals. Falconer, in his Fractal Geometry book, proves (partially) a result about the intersect of a set and a translation of a set:
Theorem 8.1. Let $E,F$ be Borel susbets of $\mathbb{R}^{n}$, then $$\dim_{H}(E\cap(F+x))\leqslant\max\{0,\dim_{H}(E\times F)-n\}$$ for almost all $x\in\mathbb{R}^{n}$.
$E\times F$ is the Cartesian product of $E$ and $F$, and $\dim_{H}$ denotes the Hausdorff dimension.
I am fine with the partial proof of it, but he goes on to say "suitable examples show that this is the best we can hope for". But he doesn't give any examples.
Does anyone know of any I can look at?