  For any finite group $G$, Thompson proved that, for each $\chi\in {\rm
Irr}(G)$, $\chi(g)$ is a root of unity or zero for more than a third of the
elements $g\in G$, and Gallagher proved that, for each larger than average
class $g^G$, $\chi(g)$ is a root of unity or zero for more than a third of the
irreducible characters $\chi\in {\rm Irr}(G)$. We show that in many cases "more
than a third" can be replaced by "more than half".
