Proportions of r-regular elements in finite classical group

TL;DR

This paper establishes optimal lower bounds on the proportion of r-regular elements in finite classical groups, independent of the field size, and also provides new upper bounds, addressing open questions in the area.

Contribution

It introduces the best possible lower bounds for r-regular elements in classical groups that do not depend on the field order, along with new upper bounds.

Findings

Lower bounds are optimal and field-independent.

New upper bounds for r-regular elements.

Answers to open questions by Pálfy and Saxl.

Abstract

For a prime $r$, we obtain lower bounds on the proportion of $r$-regular elements in classical groups and show that these lower bounds are the best possible lower bounds that do not depend on the order of the defining field. Along the way, we also provide new upper bounds and answer some open questions of the first author, P\'{a}lfy and Saxl.