ILL 1990 Problem 14

We call a set $S$ on the real line $R$ "superinvariant", if for any stretching $A$ of the set $S$ by the transformation taking $x$ to $A(x) = x_0 + a(x - x_0)$ , where $a > 0$ , there exists a transformation $B, B(x) = x + b$ , such that the images of $S$ under $A$ and $B$ agree; i.e., for any $x \in S$ , there is $y \in S$ 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.