Let $\beta \geq \alpha > 0$. Let $A\subset\mathbb R^n$ be a measurable set with Hausdorff dimension $\alpha$. Must there exist a closed connected set $B$ with Hausdorff dimension $\leq \beta$ such that $A\cap B$ has Hausdorff dimension $\alpha$? This seems like it should hold with $ \beta = \max\{\alpha ,1\}$, via some sort of ``space filling curve" type construction. I don't see a simple proof or a reference, however, and would be interested in either.
I am also interested in how regular we can take $B$ to be. For example, can we require that $B$ be the image of a curve? How about a simple curve?