On the second-largest Sylow subgroup of a finite simple group of Lie type

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large’ in the sense that $\mathrm{\mid }T{\mathrm{\mid }}^{1\mathrm{/}3}<\mathrm{\mid }S\mathrm{\mid }⩽\mathrm{\mid }T{\mathrm{\mid }}^{1\mathrm{/}2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r\mathrm{\ne }p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $\mathrm{\mid }T\mathrm{\mid }$ with $r \neq p$,theorderoftheSylow$r$-subgroupof$T$atmost$|T|^{2\lfloor\log_r(4(\ell+1) r)\rfloor/\ell}=|T|^{{\rm O}(\log_r(\ell)/\ell)}$,where$\ell$istheLierankof$T$.