“would be of size about |A|^{1/3}”. Also, my congratulations were intended for both teams, obviously. ]]>

Congratulations for a beautiful result.

]]>It seems to me that the proof you have sketched for Lemma 3 is a little more complicated than it needs to be. In particular, the use of double-sided cosets aTb is unnecessary, as is the use of the Weyl group. Emmanuel was talking to me about this; if I understood him correctly, he proceeded guided by his reading of Proposition 3.1 in my SL_3 paper (where I used the term “pivot”). The idea there was to get an element xi (the “pivot”) of a group G (acted on by another group \Upsilon) such that the map (g,y)\mapsto g\cdot y(xi) would be injective when restricted to a set A\times Y \subset G\times \Upsilon. One could obtain this pivot as follows: if there is no pivot, then A and Y are very large and the argument is easy; if there is a pivot, there has to be in some sense a last “non-pivot”, which is one step away from being a pivot under the action of A or Y; this non-pivot is shown to be in some sense constructible (or “involved”, to use your term) and thus one can construct a pivot using one more step. (As I wrote back then and as you state above, this argument can be seen as an abstraction of an argument of Konyagin and Glibichuk in the sum-product setting; what I had in SL_3 was simply the first such statement in the setting of groups.) It is clear then why Emmanuel wanted a map (g,g’)\mapsto gTg’.

In fact, this is unnecessary in this setting; one can work entirely with left cosets a T. Here’s a simpler complete proof of your Lemma 3 (which works for every simple group, just like the proof in your paper will work for every simple group).

Proof.- Suppose T’=g T g^(-1) is not involved. Look at the map phi:G\mapsto G/T’ given by g\mapsto g T’. If g and g’ map to the same element, then g^(-1) g’ lies in T’; since T’ is not involved, it must be a non-regular element of T’. Since there are at most <>|A|^{1-(dim(T)-1)/dim(G))}. Pick a set R\subset A of representatives of every preimage phi^{-1}(x), x\in phi(A).

Evidently, |R|=|phi(A)|>>|A|^{1-(dim(T)-1)/dim(G))}.

Now, because T is involved, there is a regular element x in A^(-1) A \cap T(K). Hence g x g^(-1) in A A^{-1} A A^{-1} is regular, and so |A_8\cap T(K)|>>|A|^(dim(T)/dim(G)). Now the map (g,t)->gt restricted to R\times

(A_8\times T(K)) will (a) be injective (and thus have image of size

>> |A|^{1+1/dim(G)}) and (b) have image contained in A_9. Hence |A_9|>>|A|^(1+1/dim(G)).

End of proof.

]]>An even better alternative might be to use Google Code Project Hosting (code.google.com/p). It also utilizes subversion/mercurial. Depending on how large the commit group is, it can be pretty generous (and versatile) for storage. I’ve been using it for my own projects for about a year now.

]]>Ah, good point. (I was wondering where that term came from, actually.) I’ve reworded the argument accordingly.

]]>Yes, I’ve already been informed privately by those more technologically up-to-date than I that a late 1990s era piece of software doesn’t count as “modern” nowadays, so I’m retracting that particular term :-)

For the current scale of projects (with 2-3 authors) it seems that the Subversion system works “well enough”, which is often a sufficient standard for these sorts of things. If one were to do a much larger-scale project, such as a polymath writeup, I could imagine that a distributed VCS offer further real advantages. In any case, either system is a definite advance over email-based methods.

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