# A generalization of a theorem of Rodgers and Saxl for simple groups of bounded rank

## TL;DR

This paper proves a product decomposition result for finite simple groups of Lie type with subsets of sufficiently large size, generalizing previous work and connecting to broader conjectures in group theory.

## Contribution

It extends a theorem of Rodgers and Saxl to all simple groups of Lie type with bounded rank, establishing new product decomposition conditions.

## Key findings

- Product decomposition for simple groups of Lie type
- Connection to Rodgers and Saxl's conjugacy class conjecture
- Relation to the Product Decomposition Conjecture of Liebeck, Nikolov, and Shalev

## Abstract

We prove that if $G$ is a finite simple group of Lie type and $S_1,\dots, S_k$ are subsets of $G$ satisfying $\prod_{i=1}^k|S_i|\geq|G|^c$ for some $c$ depending only on the rank of $G$, then there exist elements $g_1,\dots, g_k$ such that $G=(S_1)^{g_1}\cdots (S_k)^{g_k}$. This theorem generalizes an earlier theorem of the authors and Short. We also propose two conjectures that relate our result to one of Rodgers and Saxl pertaining to conjugacy classes in $SL_n(q)$, as well as to the Product Decomposition Conjecture of Liebeck, Nikolov and Shalev.