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 $A\_2$.



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



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> 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: $A+A+\ldots+A$.
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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.



Steinhaus theorem and Hausdorff dimension
The answer to the question is negative. Körner in [Hausdorff dimension of sums of sets with themselves](https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/188/3/90488/hausdorff-dimension-of-sums-of-sets-with-themselves) and Schmeling-Shmerkin in [On the dimension of iterated sumsets](https://arxiv.org/abs/0906.1537) showed that for any increasing sequence $\alpha\_1 < \alpha\_2 < \ldots < 1$, there is a compact set $A$, s.t. for every $k$, we have $\dim\_{H} k A = \alpha\_k$, where $kA = A + A + \ldots + A$ --- Minkowski sum of the set with itself $k$ times, and $\dim\_H$ is the Hausdorff dimension of the set. In particular, choosing any sequence such that $\dim\_H kA < 1$ for all $k$, none of the sum sets $kA$ could contain an interval.


See also the recent paper [Dimension growth for iterated sumsets](https://arxiv.org/abs/1802.03324) for some conditions under which we have dimension growth $\dim\_H (A + A) > \dim\_H(A)$. Unfortunately, it seems that the bounds there still fall short from implying that for some $k$, $\dim\_H k A = 1$, much less providing quantitative bound on $k$.


On the positive side, Marstand in "Some Fundamental Geometrical Properties of Plane Sets of Fractional Dimensions" showed that for any given pair of sets $A,B$, and almost every angle $\theta$, if we write $\lambda\_1 = \cos \theta, \lambda\_2 = \sin \theta$, then
$$ \dim\_H (A\_{\lambda\_1} + B\_{\lambda\_2}) = \min(\dim\_H(A \times B),1) \geq \min(\dim\_H(A) + \dim\_H(B), 1),$$
where $A\_{\lambda}$ is the dilatation $A\_{\lambda} := \{ \lambda x : x \in A\}$. In fact a slightly stronger statement holds: if $\dim\_H (A) = \alpha$, there is a sequence of parameters $\lambda\_1, \lambda\_2, \ldots, \lambda\_k > 0$ for $k = \lceil 1/\alpha \rceil + 1$, such that for $B := A\_{\lambda\_1} + \ldots + A\_{\lambda\_k}$ not only $\dim\_H(B)=1$, but indeed the 1-dimensional measure of this set is positive, and therefore $B+B$ contains an interval (a generic sequence of $\lambda\_i$ will do the job). Unfortunately, this result does not say anything about any specific sequence of $\lambda\_i$ (of which $\lambda\_1 = \ldots = \lambda\_k$ is the most interesting). A more modern exposition of this theorem can be found in Chapter 9 of "Geometry of sets and measures in Euclidean spaces" by Pertti Mattila.


