Derangements in non-Frobenius groups

We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$, then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is larger than $1\mathrm{/}(2n^{1\mathrm{/}2})$. This is sharp, generalizes substantially bounds of Cameron–Cohen and Guralnick–Wan, and settles conjectures of Guralnick–Tiep and Bailey–Cameron–Giudici–Royle in large degree. We also give an application to coverings of varieties over finite fields.