Denseness results for zeros and roots of unity in character tables

TL;DR

This paper investigates the distribution of zeros and roots of unity in irreducible characters of finite groups, showing that any proportion between 1/2 and 1 can be approximated arbitrarily closely by some character.

Contribution

It establishes the denseness of the set of proportions of zeros and roots of unity in irreducible characters within the interval [1/2, 1].

Findings

Any value in [1/2, 1] can be approximated by the proportion of zeros or roots of unity in some irreducible character.

The set of such proportions is dense in the interval [1/2, 1].

The result applies to all finite groups and their irreducible characters.

Abstract

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 $|\theta(\chi)-L|<\epsilon$.