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 $L\in[1/2,1]$ and any $\epsilon>0$, there exists an irreducible character $\chi$ of a finite group such that $|\theta(\chi)-L|<\epsilon$.