Denseness results for zeros and roots of unity in character tables

For any irreducible character $\chi$ of a finite group $G$, let $\theta (\chi )$ denote the proportion of elements $g\in G$ for which $\chi (g)$ is either zero or a root of unity. Then for any $L\in [1\mathrm{/}2,1]$ and any $\epsilon >0$, there exists an irreducible character $\chi$ of a finite group such that $\mathrm{\mid }\theta (\chi )-L\mathrm{\mid }<\epsilon$.