Zeros and roots of unity in character tables

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

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