E. Kowalski’s blog

It can be very rewarding to read old mathematical papers, in terms of accessing insights and ideas that may have been filtered out in later transformations of the results they contain. In my modest experience, this does not extend to notation and terminology, and it is much easier to appreciate the insights in question after translating them into modern language and formalism. This is an area where, maybe, progress is usually fairly steady. But still, there can be exceptions. I was recently rather struck, while reading the recently published collection of letters between Henri Cartan and André Weil, to discover that when they were exchanging many letters on algebraic topology just after 1945, they used the charming name cascade for what is now known as a “long exact sequence” (in homology or cohomology). I think it is too bad this didn’t become the standard name; one could have imagined that triangles

would be called “Escherian cascades”…

Incidentally, this book of letters is very interesting to read, in no small part because of the extensive notes and comments by Michèle Audin. It is published by the SMF in the same series where letters between Grothendieck and Serre were also published a few years ago.

I was last week at the conference on “Number theory and its applications” which was excellently organized by C. Delaunay and F.X. Roblot at the CIRM conference center, close to Marseille. Although I don’t have last year’s excuse at the end of the Joint Math Meetings, my remarks will be just as incoherent…

• M. Watkins showed a book he recently bought in the Canary Islands, which proves that G. Perelman is on his way to becoming a pop-culture figure:
  
  A cursory look at the content (though not by native Spanish speakers!) does not seem to suggest that this a serious work of mathematical scholarship…
• For the future writer of the definitive history of analytic number theory, I offer this remark from É. Fouvry, who said one could quote him:

  … et Chebychev arrive avec une astuce de voleur de mobylette… (…and then Chebychev comes around with a trick worthy of a bicycle thief)

• Charles Boyd, an enterprising soul worthy of homeric epithets, has ported Pari/GP to android

  

  The package can be downloaded here. There’s something confortable in having your phone factor the 8-th Fermat number during a post-dinner round-wine discussion… One may object that, at the moment, any syntax error causes the program to exit unceremoniously, but certainly this will soon improve. (Note: the broken screen was not caused by Paridroid…)

I’m currently profiting from the vacations to clean-up various aspects of my tentative to obtain explicit expansion bounds for subgroups of which are Zariski dense in . To review the previous episodes, there are three basic ingredients that needed to be made explicit:

• basic multiplicative combinatorics;
• Helfgott’s growth theorem;
• the Bourgain-Gamburd expansion criterion (especially the so-called “-flattening” lemma).

 
I’ve already talked about the first and second. For the growth theorem, after a few more changes, my current result is that any generating subset of satisfies either or

(with no condition on ).

For multiplicative combinatorics, I have reworked the argument after noticing the fact (certainly obvious to all cognoscenti) that, for the purpose I need it, one can work mainly with symmetric sets, for which the basic relation between tripling constant and growth of -fold product sets is quite a bit better (in explicit terms) than the corresponding one of “mixed” -fold products involving either a set or its inverse. This gives much better exponents.

The most important gain comes, however, from a second look at the Bourgain-Gamburd inequality. The point is that they argue from a “dyadic” viewpoint, considering (in effect) the steps of a random walk (for a suitable starting point where the walk has started to expand non-trivially). Each step from to gives a fixed improvement of the counting of closed geodesics of the corresponding length, and the number of steps which is required is directly related to the spectral gap one obtains.

From a qualitative point of view, there is nothing to argue with this. But if one wants to get explicit constants, one notices (this is also certainly known to the aforementioned cognoscenti, as shown by Helfgott’s comment on my previous post…) that the argument of Bourgain-Gamburd works essentially just as well for steps of the random walk: what is needed is a small adaptation of their main inequality to bound the spread of products where and are independent random variables with at least “as well spread out” as .

Apart from this last part, which I will include soon since I just wrote down the details, I’ve incorporated these improvements in my notes. The last point, rather satisfyingly, improves exponentially the estimates for spectral gaps: for the Lubotzky group, it becomes now , instead of (for large enough, and I confess that I don’t yet know what this last condition means explicitly…)

We’re not yet in the realm of really macroscopic numbers, but this is certainly encouraging…

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 with , 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 …]

I have one more lecture next week in my expander class, but today I finished the proof of Helfgott’s growth theorem for . As I had hoped, I did this in my notes with explicit constants (I didn’t try to follow those constants on the blackboard).

Taking into account some grains de sel, since there may well be minor computational mistakes lurking around (though I have already corrected a few), the result I obtain is the following: if is prime, and if is a symmetric generating set, containing for simplicity, then either the triple product

is all of , or otherwise we have

where

(Of course, the factor can be incorporated into a slightly-smaller exponent, but that introduces an ugly-looking dependency on the size of , which one must recover using an uglier trivial bound for small, so I preferred this version…)

The current version of the notes contains the argument, though it is a bit rough (I will soon rearrange some of it, to attempt to provide more motivation — at least the way I understand how it goes…)

For the proof, I followed the clear outline in the first sections of the paper of Pyber and Szabó. This reduces the problem, rather quickly and cleanly, to a “non-concentration” estimate for the intersection of with a regular-semisimple conjugacy class , of the type

for some fixed and absolute constant . This inequality is now commonly called a (generalized) Larsen-Pink inequality (the prototype going back to the late 90′s preprint — now published — of Larsen and Pink for the non-concentration of finite subgroups of algebraic groups in subvarieties). Though the general case is quite tricky, there is here an easy enough argument, based on studying the fibers of the map

where the three arguments are all in (this is the basic idea already presented by Larsen and Pink to explain their result, in another case).

It turns out that, if one imposes that is not the conjugacy class of elements of trace 0, which can be ensured (using bare hands) by “escape from subvarieties”, the cases where this map has positive-dimensional fibers are rather simple to analyze (I owe this computation to R. Pink…)

Moreover, only one case requires another instance of Larsen-Pink-type inequalities (those readers who have looked at the paper of Larsen and Pink, or the one of Breuillard-Green-Tao which has a general “approximate” version, will know that there is a rather complicated induction involved in general), and it is a very easy one: if is the subgroup of upper-triangular unipotent matrices, then

which is an instructive exercise. (In fact, in rearranging this section of my notes, I will use this as a motivating example…)

With this final ingredient, I can now produce (with the same amount of salt…) an effective spectral gap for the Cayley graphs of the Lubotzky subgroup of , generated by

modulo primes, namely (drumroll) for large enough (drumroll; but I won’t tell you how large today), we have (drumroll)

(Actually, I already know various points of inefficiency in my treatment of the Bourgain-Gamburd expansion argument which should lead to some improvements, and I hope to find other avenues to explore and stones to turn to do better…)