A group is $d$-quasirandom if every nontrivial complex representation has dimension at least $d$. Gowers introduced quasirandomness in [this paper](https://arxiv.org/abs/0710.3877) and proved that every nonabelian finite simple group of order $n$ is $\sqrt{\log n}/2$-quasirandom.


**Question:** What is the correct (asymptotic) lower bound for the quasirandomness of nonabelian finite simple groups?


Gowers indicates that a theorem of Jordan (Theorem 14.12 [here](https://www.cefns.nau.edu/%7Efalk/classes/511/Isaacs_Character_theory.pdf)) implies a slightly better bound. Indeed, this theorem gives that a nonabelian simple group of order $n$ has a nontrivial representation of dimension $d$ only if $n\leq (d!)12^{d(\pi(d+1)+1)}$, which implies that these groups are $\Omega(\sqrt{\log n\log\log n})$-quasirandom.


Of course, the correct bound should be obtainable using the classification of finite simple groups. The sporadic groups are irrelevant to the asymptotics, and the alternating groups are $\Omega(\log n/\log\log n)$-quasirandom. It remains to study the groups of Lie type. What I want can be read off the first column of the generic character tables of these groups, but to my surprise, these aren't known yet (see LeechLattice's answer [here](https://mathoverflow.net/questions/391136/is-the-representation-of-finite-simple-groups-fully-understood/)). Interestingly, the answer [here](https://mathoverflow.net/questions/400864/is-operatornamepsl2-q-the-most-quasirandom-group) describes how to find an upper bound on the smallest nontrivial representation in these cases, whereas I want a lower bound.



How quasirandom are the nonabelian finite simple groups?
Suppose $G$ is a finite simple group of order $n$ with a nontrivial representation of degree $d$. Then $G$ is isomorphic to a subgroup of $U(d)$. By Collins's sharp version of Jordan's theorem (<https://www.degruyter.com/document/doi/10.1515/JGT.2007.032/html>), $G$ has an abelian normal subgroup of index at most $(d+1)!$, which must be trivial since $G$ is simple, so $|G| \leq (d+1)!$. Rearranging, $d \gtrsim \log n / \log\log n$.


Collins's work builds on work of Weisfeiler that I think was unfinished by the time of his disappearance.


**Edit:** Specializing to simple $G$ actually reduces Collins's paper to a reference to a reference to the paper of Seitz and Zalesskii (<https://www.sciencedirect.com/science/article/pii/S0021869383711324?via%3Dihub>) mentioned by David Craven in the comments, so that's really the heart of the matter. We thereby get the slightly stronger bound $n \leq (d+1)!/2$ (for sufficiently large $d$). Apart from alternating groups I think you can read out a much stronger bound like $n \leq \exp O((\log d)^2)$, or $d \geq \exp \Omega( (\log n)^{1/2})$ (I am guessing the next worst case is $\mathrm{SL}\_n(2)$).


