TITLE: A compact set in the plane with small sum-set and large projections
QUESTION [5 upvotes]: 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.

REPLY [3 votes]: 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 $\cap_k F_k$).