# Research Updates: Boston–Shalev for conjugacy classes, growth in linear groups, and the (amazing) Kelley–Meka result

1. Boston–Shalev for conjugacy classes

Last week Daniele Garzoni and I uploaded to the arxiv a preprint on the Boston–Shalev conjecture for the conjugacy class weighting. The Boston–Shalev conjecture in its original form predicts that, in any finite simple group $G$, in any transitive action, the proportion of elements acting as derangements is at least some universal constant $c > 0$. This conjecture was proved by Fulman and Guralnick in a long series of papers. Daniele and I looked at conjugacy classes instead, and we found an analogous result to be true: the proportion of conjugacy classes containing derangements is at least some universal constant $c' > 0$.

Our proof depends on the correspondence between semisimple conjugacy classes in a group of Lie type and polynomials over a finite field possibly with certain restrictions: either symmetry or conjugate-symmetry. We studied these sets of polynomials from an “anatomical” perspective, and we needed to prove several nontrivial estimates, e.g., for

• the number of polynomials with a factor of a given degree (which is closely related the “multiplication table problem”),
• the number of polynomials with an even or odd number of irreducible factors,
• the number of polynomials with no factors of small degree,
• or the number of polynomials factorizing in a certain way (e.g., as $f = gg^*$, $g$ irreducible, $g^*$ the reciprocal polynomial).

For a particularly neat example, we found that, if the order of the ground field is odd, exactly half the self-reciprocal polynomials have an even number of irreducible factors — is there a simple proof of this fact?

2. Growth in Linear Groups

Yesterday Brendan Murphy, Endre Szabo, Laci Pyber, and I uploaded a substantial update to our preprint Growth in Linear Groups, in which we prove one general form of the “Helfgott–Lindenstrauss conjecture”. This conjecture asserts that if a symmetric subset $A$ of a general linear group $\mathrm{GL}_n(F)$ ($n$ bounded, $F$ an arbitrary field) exhibits bounded tripling, $|A^3| \le K|A|$, then $A$ suffers a precise structure: there are subgroup $H \trianglelefteq \Gamma \le \langle A \rangle$ such that $\Gamma / H$ is nilpotent of class at most $n-1$, $H$ is contained in a bounded power $A^{O_n(1)}$, and $A$ is covered by $K^{O_n(1)}$ cosets of $\Gamma$. Following prodding by the referee and others, we put a lot more work in and proved one additional property: $\Gamma$ can be taken to be normal in $\langle A \rangle$. This seemingly technical additional point is actually very subtle, and I strongly doubted whether it was true late into the project, more-or-less until we actually proved it.

We also added another significant “application”. This is not exactly an application of the result, but rather of the same toolkit. We showed that if $G \le \mathrm{GL}_n(F)$ (again $F$ an arbitrary field) is any finite subgroup which is $K(n)$-quasirandom, for some quantity $K(n)$ depending only on $n$, then the diameter of any Cayley graph of $G$ is polylogarithmic in the order of $|G|$ (that is, Babai’s conjecture holds for $G$). This was previously known for $G$ simple (Breuillard–Green–Tao, Pyber–Szabo, 2010). Our result establishes that it is only necessary that $G$ is sufficiently quasirandom. (There is a strong trend in asymptotic group theory of weakening results requiring simplicity to only requiring quasirandomness.)

The intention of our paper is more-or-less to “polish off” the theory of growth in bounded rank. By contrast, growth in high-rank simple groups is still poorly understood.

3. The Kelley–Meka result

Not my own work, but it cannot go unmentioned. There was a spectacular breakthrough in additive combinatorics last week. Kelley and Meka proved a Behrend-like upper bound for the density of a subset $A \subset \{1, \dots, n\}$ free of three-term arithmetic progressions (Roth’s theorem): the density of $A$ is bounded by $\exp(-c (\log n)^\beta)$ for some constants $c, \beta > 0$. Already there are other expositions of the method which are also worth looking at: see the notes by Bloom and Sisask and Green (to appear, possibly).

Until this work, density $1 /\log n$ was the “logarithmic barrier”, only very recently and barely overcome by Bloom and Sisask. Now that the logarithmic barrier has been completely smashed, it seems inevitable that the new barometer for progress on Roth’s theorem is the exponent $\beta$. Kelley and Meka obtain $\beta = 1/11$, while the Behrend construction shows $\beta \le 1/2$.

# The apparent structure of dense Sidon sets

“What are dense Sidon sets of {1, …, n} like?”, asked Tim Gowers on his blog almost ten years ago. A Sidon set is a set without any solutions to $x+y=z+w$, which in additive combinatorics jargon means that it has minimal additive structure. Almost paradoxically, large sets with this property appear to be structured in another way, and that’s a bit of a mystery currently.

Now Freddie Manners and I have an idea about the answer to Gowers’s question, at least if “dense” means “really really dense, like 99% as dense as possible”, and the setting is a finite abelian group like $\mathbf{Z}/n\mathbf{Z}$ instead of $\{1, \dots, n\}$. Our suspicion is that any such set must be related in a specific way to the collineation group of a finite projective plane.

We uploaded our paper to the arxiv today, so have a look and tell us what you think! I also spoke about this recently at CANT 2021. The recording is available here: https://youtu.be/s4ItIkkUvF4

On the other hand, there is also some evidence pointing the other way. Forey and Kowalski showed recently that certain moderately dense Sidon sets arise from algebraic geometry, not from projective planes. It is not clear whether such sets reach the “really really dense” threshold; if so, that would contradict our conjecture, but I suspect they don’t.