**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](https://arxiv.org/pdf/1805.01997) imply the affirmative answer to this problem for compact subsets $K$ of positive dimension in the plane.



A compact set in the plane with small sum-set and large projections
The answer is negative. The key point is the following


**Lemma:** For every $L>0$ and $x\in\mathbb R^2, x\ne 0$, there exist $\ell>0$ and a closed origin-symmetric set $F\subset\mathbb R^2$ such that every interval $I$ of length $L$ on the plane contains a subinterval $J\subset I\cap F$ of length $\ell$ while $F\cap(F+x)=\varnothing$.


**Proof:** WLOG, $x=(0,1)$. Now choose a huge integer $N=N(L)$ and consider the sinusoidal strip $S=\{(x,y):|y-\sin x|\le\frac 1{10(2N+1)}\}$. Take $F=\cup\_{k\in\mathbb Z}(S+(0,\frac{2k}{2N+1}))$.


Now just enumerate some dense set $x\_1,x\_2,\dots$ on the plane, take $L\_1=1$ and use $x\_k$ and $L\_k$ to construct $F\_k$ and $L\_{k+1}$. Put $K=\cap\_k F\_k\cap \bar D(0,10)$, say (note that each finite intersection of $F\_k$ intersects every interval of length $1$, so this property will survive for the full intersection $\cap\_k F\_k$).


