We call a set $ S$ on the real line $ \mathbb{R}$ [i]superinvariant[/i] if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) \equal{} x_0 \plus{} a(x \minus{} x_0), a > 0$ there exists a translation $ B,$ $ B(x) \equal{} x\plus{}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) \equal{} B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) \equal{} A(u).$ Determine all [i]superinvariant[/i] sets.
We need to determine all sets \( S \subseteq \mathbb{R} \) that are superinvariant. A set \( S \) is superinvariant if for any stretching transformation \( A(x) = x_0 + a(x - x_0) \) with \( a > 0 \), there exists a translation \( B(x) = x + b \) such that the images of \( S \) under \( A \) and \( B \) agree.

We claim that the superinvariant sets are \( \varnothing \), \( \mathbb{R} \), \( \{p\} \), \( \mathbb{R} \setminus \{p\} \), \( (p, \infty) \), \( [p, \infty) \), \( (-\infty, p) \), and \( (-\infty, p] \).

1. **Verification of the claimed sets:**
   - **Empty set \( \varnothing \)**: Clearly, any transformation of the empty set remains the empty set.
   - **Entire real line \( \mathbb{R} \)**: Any stretching or translation of \( \mathbb{R} \) is still \( \mathbb{R} \).
   - **Single point \( \{p\} \)**: Any stretching or translation of a single point \( p \) remains a single point.
   - **Complement of a single point \( \mathbb{R} \setminus \{p\} \)**: Any stretching or translation of \( \mathbb{R} \setminus \{p\} \) remains \( \mathbb{R} \setminus \{p\} \).
   - **Intervals \( (p, \infty) \), \( [p, \infty) \), \( (-\infty, p) \), and \( (-\infty, p] \)**: These intervals are preserved under stretching and translation.

2. **Conversely, suppose \( S \) is a superinvariant set. We consider two cases:**

   **Case A: \( S \) is periodic.**
   - If \( S \) is periodic, there exists a period \( t > 0 \) such that \( S + t = S \).
   - If \( t \) is a period, then scaling \( S \) by a factor of \( a \) shows that \( at \) is also a period of \( S \).
   - Hence, either \( S \) is empty, or else we can pick \( s \in S \) and then deduce that \( s - at, s + at \in S \) for all \( a > 0 \). By varying \( a \), we see that \( S = \mathbb{R} \).

   **Case B: \( S \) is not periodic.**
   - For a given stretching \( A \), we have a unique choice of the constant \( b \) in the translation \( B \).
   - Let \( a_1, a_2 \) be positive reals not equal to 1. There exist \( b_1, b_2 \in \mathbb{R} \) such that \( S \) is invariant under the transformations \( A_1(x) = a_1 x - b_1 \) and \( A_2(x) = a_2 x - b_2 \).
   - Applying \( A_1 \) and then \( A_2 \) gives the transformation \( x \mapsto a_1 a_2 x - (a_2 b_1 + b_2) \).
   - Applying \( A_2 \) and then \( A_1 \) gives the transformation \( x \mapsto a_1 a_2 x - (a_1 b_2 + b_1) \).
   - Since \( S \) is invariant under both transformations, we have \( a_1 b_2 + b_1 = a_2 b_1 + b_2 \), i.e., \( \frac{b_1}{b_2} = \frac{a_1 - 1}{a_2 - 1} \).
   - By varying \( a_1 \) and \( a_2 \), we can find a constant \( p \) such that for any \( a > 0 \), \( S \) is invariant under \( A(x) = ax - p(a - 1) \), i.e., \( A(x) = p + a(x - p) \).
   - This implies that any stretching about \( p \) preserves \( S \).

3. **From this, we deduce:**
   - If \( S \) contains any point \( q > p \), then \( S \) contains the entire interval \( (p, \infty) \).
   - If \( S \) contains any point \( q < p \), then \( S \) contains the entire interval \( (-\infty, p) \).

Combining these observations, we conclude that \( S \) is a union of some of the following sets: \( (-\infty, p) \), \( \{p\} \), \( (p, \infty) \), which gives us the answers.

The final answer is \( \boxed{ \varnothing } \), \( \mathbb{R} \), \( \{p\} \), \( \mathbb{R} \setminus \{p\} \), \( (p, \infty) \), \( [p, \infty) \), \( (-\infty, p) \), \( (-\infty, p] \).