Many zeros of many characters of GL(n,q)

Abstract

For $G=\mathrm{GL}(n,q)$, the proportion $P_{n,q}$ of pairs $(\chi ,g)$ in $\mathrm{Irr}(G)\times G$ with $\chi (g)\ne 0$ satisfies $P_{n,q}\to 0$ as $n\to \infty$.