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.

• 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…)

First, a raptor looking at me straight in the eyes,

and then a shingleback lizard from the Zürich Zoo being handed his lunch on chopsticks:

he (or she) was very lazy about actually starting eating his (or her) cricket, which makes you wonder how things would go in the wild…

And finally a leaf-fish, still from the Zürich Zoo, which is a species I had never seen before:

• 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…

(originally available for MS-DOS and Windows 3.1) as well as the American Heritage Dictionary

(though I use the O.E.D instead when I’m connected to the ETH network). The Grand Robert is the best anti-pedant tool I know against so-called défenseurs de la langue française; it usually reveals that their favorite anglicisms are perfectly French (e.g., opportunité, in the sense of “occasion, circumstance”, which goes back to 1355 in French, and is at least as French as Baudelaire…)

I’m even more impressed to be able to boot the equivalent of my old Mac SE30,

and thereby play with, or recover, the old files I used to work with during my PhD thesis and before. (In fact, the emulator boots in something like 1.5 seconds on my laptop, which is about a hundred times faster than it ever did in real life…) Afficionados will note the realistic 512 x 384 resolution of the screen.

[P.S. For the Lubotzky group, the spectral gap goes down to something like …]

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…)

or a cycle of length 2:

The first case occurs when one is not careful, taking as edge set of the set

(as was probably to be expected, my notes on expander graphs use this naïve definition…)

The second case occurs (at least) when one is Serre, as can be checked from the definition in “Trees”, and the resulting illustration:

(in a French printing of that book which I have seen, Serre specifically warns that the two edges are not the same).

The questions are, which is the “right” definition, if any? and should this matter? To a large extent, the answer to the second is (fortunately) “not really”, though the same issue arises for any Cayley graph where some generator is an involution.

Either choice has some potentially annoying features:

(1) The canonical choice (Serre’s) has the effect that any Cayley graph of a group with respect to a generating set containing an involution has multiple edges; for instance, the following picture taken from P. de la Harpe’s “Topics in geometric group theory”

is supposed to be the Cayley graph of with respect to a generating set for which it is a free product of cyclic groups of order 2 and 3 (generated by and , respectively), but is wrong with the canonical definition. (To be fair, de la Harpe mentions the issue, and says that he uses the naïve definition because that is not a problem for his purposes.)

(2) In the canonical choice, the formula

for the Markov averaging operator is usually wrong. (E.g., for the example of above.) In particular, the spectrum of the Laplace operator is not the same depending on which definition is used…

(3) Similarly, the Cayley graph is not always -regular.

(4) However, in the non-canonical choice, it follows that a Cayley graph of a finite group may have infinite girth (being a tree, as in the example of …) Correspondingly, the interpretation of the girth of a Cayley graph as the smallest length of a non-trivial relation in the generators may fail…

At the moment, as I said, I use the naïve definition in my notes (most books I’ve seen which bother to define the Cayley graph in a formally precise way seem to do the same, though the definition is often sufficiently fuzzy that it doesn’t take much effort to accommodate both possibilities…) But I claim somewhere that the Cayley graph of (with respect to ) is a cycle of length , which is then wrong for .

I haven’t quite decided how I will change the text, since I tend to like the formula for the Markov averaging operator. Maybe, like de la Harpe, I will just mention both possibilities and use the naïve one. Certainly, as far as expander graphs are concerned, this has no consequence on the property of being or not an expander for a sequence of Cayley graphs (of bounded valency).

The first thing to do in order to obtain an explicit estimate is to get an explicit form of the relation between “pairs of sets with large multiplicative energy” and “cosets of a common approximate group” — indeed, this is a crucial point in the first step of the argument discovered by Bourgain and Gamburd to derive expansion for some Cayley graphs from a classification of approximate subgroups (or more directly, of sets with “small tripling”, a classification which had been produced by Helfgott for ).

To explain this, recall that if are subsets of a finite group , the normalized multiplicative energy is defined by

It is easy to show that , and it is a pleasant exercise to prove that the extreme case occurs if and only if there exists a subgroup of and elements , of such that

The Bourgain-Gamburd argument depends on understanding sets and such that

for some (small) .
This can now be done, in some cases, by first proving an “approximate” version of the characterization of the extreme , and then classifying the resulting “approximate” objects. The second step is much more involved, and I won’t talk about it here, but the first can be done in full generality. The standard texts explaining this are the book of Tao and Vu and a paper of Tao (which contains more details than the summary in the book.)

It is clear from reading the proofs that they are “effective”, but these sources do not give explicit inequalities. So I did the exercise of going through the arguments to get actual constants, inputing the recent explicit results of Petridis to get better control on product sets than the corresponding argument in Tao’s paper. After three or four rounds of checks, I get the following: if

then there exist a -approximate subgroup of and elements , ,
such that

where

(and a -approximate subgroup is defined to be a symmetric subset, containing the identity, such that for some subset of size .)

For the moment, I’ve only typed a very condensed note spelling this out, which is found on my page of unpublished notes; I admit that I had some fun devising an arrow notation to simplify keeping track of the relation and sizes between the sets involved…

All this will be incorporated in my lectures notes on expanders, and then expanded later to a full proof so that the latter are self-contained. I will also attempt to improve these constants, which are not very promising when I think of what the spectral gap will become in the end…

(Interestingly, according to Wikipedia, McCarthy had a PhD in math under Lefschetz…)