  Suppose $E$ is a subset of the unit circle $\mathbb{T}$ and $H^\infty\subset
L^\infty$ is the Hardy subalgebra. We examine the problem of finding the
distance from the characteristic function of $E$ to $z^nH^\infty$. This admits
an alternate description as a dual extremal problem. Precise solutions are
given in several important cases. The techniques used involve the theory of
Toeplitz and Hankel operators as well as the construction of certain conformal
mappings.
