# And then there was one in 744

The feared grains of salt have reared their ugly heads: the growth exponent 1/186 that I mentioned in my last post has been reduced to 1/744. I had missed the fact that, in order to avoid dealing with elements of trace 0, I first had to replace the generating set $H$ with $H\cdot H$, alas… But the previous exponent does work for any generating set which contains a regular semisimple element with non-zero trace.

[P.S. For the Lubotzky group, the spectral gap goes down to something like $2^{-2^{36}}$…]

### Kowalski

I am a professor of mathematics at ETH Zürich since 2008.

## 4 thoughts on “And then there was one in 744”

1. I can’t help of thinking of the fourier coefficients of the j-invariant. (but this could just be a random coincidence too)

1. I am pretty certain this is just a coincidence…

2. Emmanuel,

>in order to avoid dealing with elements of trace 0, I >first had to replace the generating set H with H\cdot >H, alas…

Surely, whether your goal is to get a bound on the diameter or a bound on the spectral gap, it is best to prove a theorem of the following form?

Let A, B be sets of generators of G=SL_2(F_p). Then
either

|BBABB^{-1}BBABBBABB| > |A|^{1+\delta}

or

(same word) = G.

I just made that word up – it’s just an example. The point is that you could use a word with many occurences of B and few occurences of A. That way, if you are bounding the diameter of the Cayley graph Gamma(G,S) (say), you can initially set A = S, B = S, and then A = word(S,S), B = S, and then
A = word(word(S,S),S), B = S…
In other words, one of the sets is kept small and cheap – and is used to escape from special situations (such as zero trace).

Also notice that the tripling Lemma (Ruzsa) does involve a certain cost, so, if you care about constants, you are better off not using it if at all possible.

1. That’s certainly the right approach indeed… I had already started implementing this idea, for the expansion bound à la Bourgain-Gamburd, since it’s in fact the least efficient part of the argument when it comes to explicit bounds. I’ll report soon on what I can get now…