A compact set in the plane with small sum-set and large projections

Problem. Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane?

Remark. The results of this paper imply the affirmative answer to this problem for compact subsets $K$ of positive dimension in the plane.

• @David Thank you for your interest to this question. For positive dimension the argument is as follows. If a compact set $K$ of the plane has positive dimension, then $K$ contains a non-trivial continuum $C$. If this continuum is not contained in a line, then we apply Theorem 2 of (arxiv.org/pdf/1805.01997.pdf) and conclude that $C+C$ and $C-C$ has non-empty interior in the plane. It remains to consider the case when all connected components of $K$ are contained in affine lines. – Taras Banakh May 15 '18 at 22:22
• @David If two non-trivial connected components are contained in non-parallel lines then their sum and difference have non-empty interior and we are done. So, it remains to consider the case when all connected components of $K$ lie in parallel lines. In this case fix any non-trivial connected component $C$ of $K$ and consider the projection of $K$ onto a line $l$ orthogonal to $C$. Since the projection of $K$ on $l$ has non-empty interior, there exists a subset $D$ of $K$ whose diameter is much smaller than the diameter of $C$ and the projectionof $D$ onto $l$ still has non-empty interior. – Taras Banakh May 15 '18 at 22:28
• @David It is easy to see that the sets $D+C$ and $D-C$ have non-empty interiors in the plane. – Taras Banakh May 15 '18 at 22:31