  If a Jacobi matrix $J$ is reflectionless on $(-2,2)$ and has a single
$a_{n_0}$ equal to 1, then $J$ is the free Jacobi matrix $a_n\equiv 1$,
$b_n\equiv 0$. I'll discuss this result and its generalization to arbitrary
sets and present several applications, including the following: if a Jacobi
matrix has some portion of its $a_n$'s close to 1, then one assumption in the
Denisov-Rakhmanov Theorem can be dropped.
