On the proportion of vanishing elements in finite groups

TL;DR

This paper investigates the distribution of elements in finite groups that are zeros of irreducible characters, revealing that their proportion takes on very sparse values across a large part of the interval from 0 to 1.

Contribution

It establishes that the function measuring the proportion of zeros of irreducible characters in finite groups has very sparse values over a large segment of the interval.

Findings

The function $ ext{P}_v(G)$ takes sparse values in most of the interval [0,1].

Zeros of irreducible characters are rare in certain proportions.

The results provide new insights into the structure of finite groups and their character theory.

Abstract

We prove that the function $\mathrm{P}_{\mathrm{v}}(G)$, measuring the proportion of the elements of a finite group $G$ that are zeros of irreducible characters of $G$, takes very sparse values in a large segment of the $[0,1]$ interval.