I put an answer (due to Blichfeldt, not me) to essentially this question at your earlier question. To address the problem raised by Richard Stanley, one result I know in this direction is by John Thompson: if $\chi$ is an irreducible character of a finite group $G$, then there are more than $|G|/3$ elements at which the value taken by $\chi$ are is either zero or a root of unity.
I put an answer (due to Blichfeldt, not me) to essentially this question at your earlier question. To address the problem raised by Richard Stanley, one result I know in this direction is by John Thompson: if $\chi$ is an irreducible character of a finite group $G$, then there are more than $|G|/3$ elements at which the value taken by $\chi$ are either zero or a root of unity.