Donald E. Marshall and Carl Sundberg
\def\om{\omega} \def\b{\partial} \def\D{{\Bbb D}} A powerful tool for studying the growth of analytic and harmonic functions is Hall's Lemma, which states that there is a constant $C>0$ so that the harmonic measure of a subset $E$ of the closed unit disk $\overline\D$ evaluated at $0$ satisfies $$\om(0,E,\D\setminus E) \ge C \om(0,E_{rad},\D),\eqno(*)$$ where $E_{rad}$ is the radial projection of $E$ onto $\b\D$. FitzGerald, Rodin and Warschawski proved that if $E$ is a continuum in $\overline \D$ whose radial projection has length at most $\pi$ then (*) is true with $C=1$, and they asked how large the length, $|E_{rad}|$, can be in order for their result to be valid. We prove that (*) holds with $C=1$ provided $|E_{rad}|\le \theta_c \simeq 2\pi\left({{350}\over{360}}\right)$ and $\theta_c$ cannot be replaced by a larger number. Fuchs asked for the largest constant $C$ so that (*) holds for all $E$. We show that for every continuum $E\subset \overline\D$, (*) holds with $C=C_{2\pi}\simeq .977126698498665669\dots$, where $C_{2\pi}$ is the harmonic measure of the two long sides of a 3:1 rectangle evaluated at the center. There are Jordan curves for which equality holds in (*) with $C=C_{2\pi}$.