I have found a theorem which is proven for the base case but I would like to try and proof the entire thing. I am hoping someone could help me by pointing the way forward.
Notation. $\dim_{H}$ is the Hausdorff dimension: https://en.wikipedia.org/wiki/Hausdorff_dimension.
Theorem. Let $E,F\subset\mathbb{R}^{n}$ be Borels sets. Then, for almost all $x\in\mathbb{R}^{n}$, $\dim_{H}(E\cap(F+x))\leqslant\max\{0,\dim_{H}(E\times F)-n\}$.
I am given that, for a Borel subset $F$ of $\mathbb{R}^{2}$, for almost all $x$, \begin{equation}\tag{1}\label{1} \dim_{H}(F\cap L_{x})\leqslant\max\{0,\dim_{H}(F)-1\}, \end{equation} where $L_{x}$ is the line parallel to the $y$-axis through the point $(x,0)$.
Proof (when $n=1$). Let $L_{c}$ be the line in the $(x,y)$-plane with equation $x=y+c$. Assuming $\dim_{H}(E\times F)>1$ (why can we do that?), it follows from \eqref{1} that \begin{equation}\tag{2}\label{2} \dim_{H}((E\times F)\cap L_{c})\leq\dim_{H}(E\times F)-1 \end{equation} for almost all $c\in\mathbb{R}$. But at a point $(x,x+c)\in (E\times F)\cap L_{c}$ if and only if $x\in E\cap(F+c)$. So, for each $c$, the projection onto the $x$-axis of $(E\times F)\cap L_{c}$ is the set $E\cap(F+c)$. In particular, $\dim_{H}(E\cap(F+c))=\dim_{H}((E\times F)\cap L_{c})$, so the result follows from \eqref{2}.
I understand this proof (except the bracketed part), just not how to generalise it to higher dimensions.