# Character Levels and Character Bounds. II

## TL;DR

This paper extends a level theory and character bounds from simple groups of linear and unitary type to all classical groups, improving previous results and providing stronger bounds for elements with small centralizers.

## Contribution

It generalizes and strengthens previous character bounds to include all classical groups, broadening the scope of the original level theory.

## Key findings

- Extended character bounds to all classical groups.
- Strengthened bounds for elements with small centralizers.
- Unified approach for linear, unitary, and classical groups.

## Abstract

This paper is a continuation of [GLT], which develops a level theory and establishes strong character bounds for finite simple groups of linear and unitary type in the case that the centralizer of the element has small order compared to $|G|$ in a logarithmic sense. We strengthen the results of [GLT] and extend them to all groups of classical type.