Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem and it easily follows by looking at neighborhoods of a density point of $A_1$ and a density point of $A_2$.

Question. Let $\alpha\in (0,1)$. Is it true that then there is $k$ such that if $\mathcal{H}^\alpha(A_i)>0$, $i=1,2,\ldots,k$, then $A_1+A_2+\ldots+A_k$ contains an interval?

Here $\mathcal{H}^{\alpha}$ stands for the Hausdorff measure.

I am also interested in a version of this question where $A_1=A_2=\ldots=A_k$ i.e., if we add a set $A$ with $\mathcal{H}^\alpha(A)>0$ to itself $k$ times: $A+A+\ldots+A$.

If this is not true, but simiar results are true, I would be interested in references. I am particulary interested in sets in a line.