Question 29
(FIN 2) We call a set $S$ on the real line $\mathbb R$ superinvariant if for any stretching $A$ of the set by the transformation taking $x$ to $A(x)=x_0+a(x-x_0)$ there exists a translation $B$, $B(x)=x+b$, such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x\in S$ there is a $y\in S$ such that $A(x)=B(y)$ and for any $t\in S$ there is a $u\in S$ such that $B(t)=A(u)$. Determine all superinvariant sets.
Remark. It is assumed that $a>0$.
Remark. It is assumed that $a>0$.