Zeros and roots of unity in character tables

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