Timeline for A categorical framework for Freiman s-morphisms
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| Sep 26, 2021 at 1:00 | comment | added | David Roberts♦ | The updated link for Terry Tao's paper Product set estimates for non-commutative groups (linked in this comment) is arxiv.org/abs/math/0601431 | |
| Oct 18, 2012 at 0:41 | comment | added | Salvo Tringali | Indeed, my question originates from Hegyvari and Hennecart's paper: hegyvari.web.elte.hu/Freimanmodels.pdf, which in turn is inspired by Green and Ruzsa's work. I've given a read of Tao's paper. It is definitely focused on measure-theoretic/metric/entropic analogues of (many) results from the theory of sum-sets but still in the (non-commutative) setting of groups (as far as I'm concerned, I don't like much the notational distinction between the commutative and non-commutative case). Freiman morphisms seem to enter the scene in Sect. 7, but they are not even mentioned explicitly. | |
| Oct 17, 2012 at 11:52 | comment | added | user9072 | Some more references: s-models (finite, minimal) by Green and Ruzsa arxiv.org/abs/math.NT/0505198 (commutative!) ; for non-commutative see eg last entry on the notes page of Green dpmms.cam.ac.uk/~bjg23/notes.html . A paper in the non-commutative case containing also quite a bit foundational material (in the sense of fixing/harmonising various notions) is I think Tao's Prod. set estim. for n-comm. grps front.math.ucdavis.edu/math.CO/0601431 For the added names: of course (also I only had some quite specific developpment in mind, not a general list for AddComb). | |
| Oct 16, 2012 at 20:02 | comment | added | Salvo Tringali | @quid. In all cases, thank you much for the fruitful conversation. I won't accept your answer only because I would like to hear other voices on the same issue, and possibly some categorist. Lastly, I must really add N. Hegyvari and I. Ruzsa to the list of the people working on the subject. I feel it as necessary. :) | |
| Oct 16, 2012 at 19:40 | comment | added | Salvo Tringali | You have nothing to be sorry about! As a minor remark, there is no loss of generality in assuming that the $\varepsilon_i$'s are all equal to 1, as far as the abelian case is concerned. I deliberately phrased the whole stuff in the form that I did to highlight that everything goes through verbatim in the non-commutative setting. On another hand, it is somewhat apparent, I think, that the true problems arise when trying to develop these (basic) ideas further and approach the question from the top (and not from the bottom). In any case, I'm happy to know that this has good chances to be new. | |
| Oct 16, 2012 at 19:14 | history | edited | user9072 | CC BY-SA 3.0 |
deleted some 'vague comment' as it appears to be wrong.
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| Oct 16, 2012 at 19:13 | comment | added | user9072 | Yes, I mixed up something (yet not even sure what exactly). Sorry about that. The problem was that I believed to remember (in error it seems) that there are issues with composition. [Possibly I confused this with things like if A,B s-F-iso then nA and nB (s/n)-F-iso where the order of the iso/hom changes.] I delete(d) the 'vague comment'. | |
| Oct 16, 2012 at 19:12 | comment | added | Salvo Tringali | a $s$-morphism. For if $x_1, \ldots, x_s, y_1, \ldots, y_s \in X_1$ and $\varepsilon_1, \ldots, \varepsilon_s \in \{\pm 1\}$ are such that $\sum_i \varepsilon_i x_i=\sum_i \varepsilon_i y_i$ then $\sum_i \varepsilon_i\phi(x_i) =\sum_i \varepsilon_i\phi(y_i)$ (since $\phi$ is a $s$-morphism from $(\mathfrak A_1, X_1)$ to $(\mathfrak A_2, X_2)$), to the effect that $\sum_i\varepsilon_i\psi(\phi(x_i))=\sum_i\varepsilon_i\psi(\phi(y_i))$ (since $\phi(x_i),\phi(y_i)\in Y_1$ and $\pi$ is a $s$-morphism from $(\mathfrak B_1,Y_1)$ to $(\mathfrak B_2,Y_2)$. | |
| Oct 16, 2012 at 19:00 | comment | added | Salvo Tringali | $X_i \subseteq \mathfrak A_i$ and $\phi: X_1 \to X_2$ a function (I'm slightly abusing notation, but it's not a great issue), such that blah blah blah. Thus, I don't see any problem here: If $\phi: (\mathfrak A_1, X_1) \to (\mathfrak A_2, X_2)$ and $\psi: (\mathfrak B_1, Y_1) \to (\mathfrak B_2, Y_2)$ are two $s$-morphisms, one defines the composition of $\phi$ with $\psi$ iff $(\mathfrak A_2, X_2) = (\mathfrak B_1, Y_1)$, as expected, accordingly setting it equal to the arrow $h: (\mathfrak A_1, X_1) \to (\mathfrak B_2, Y_2)$ for which $h = \psi \circ_{\bf Set} \phi$. This is still (...) | |
| Oct 16, 2012 at 18:46 | comment | added | Salvo Tringali | Thank you once more for your (rich) comments. Indeed, I've just realized that also Tao and Vu work out only the abelian case (I had forgot that they speak of additive groups wherever commutativity is implied). As for the mathematical question that you pose, I don't see a problem here. It really depends on the way you're defining your category. Though it doesn't use explicitly this wording Additive Combinatorics defines - let me say, very appropriately - a Freiman $s$-morphism as an arrow $\phi: (\mathfrak A_1, X_1) \to (\mathfrak A_2, X_2)$, with $\mathfrak A_i$ a (commutative) group (...) | |
| Oct 16, 2012 at 18:17 | comment | added | user9072 | (...) But I certainly do not want to claim that doing such foundational work you envision would be 'not interesting'. Likely it can be very interesting in its own right and of course could have interesting applications. This is also what I tried to convey with the 'left to explore'. As said, in particular in the non-commutative setting I think quite little was done (yet), and so a lot remains to be done by somebody. You perhaps. So, in brief, my answer is: categorically, nothing AFAIK. But there are some general considerations. So to further expand them could be interesting. | |
| Oct 16, 2012 at 18:09 | comment | added | user9072 | (...) So also the books are not written in a very 'structural' style. The notion is needed at some point so it is introduced and developped to the point it seems useful for the things at hand. If later generalizations should be needed, they can be done later. (Oversimplifying a lot.) Also until recently this was a very small subject. So people did what they found most interesting and useful, which did not so much include abstarct foundational considerations (for all I know), and as I tried to convey presumably also for reasons of personal style and taste. (...) | |
| Oct 16, 2012 at 17:59 | comment | added | user9072 | You are welcome. Yes this subject is (or rather was until quite/very recently) very commutative. There is some classical non-commutative work (eg by Freiman, Olson&White) but the focus was commutative; it is even called Additive Combinatorics after all, and this name is rather recent (Tao 2003, I'd guess). The developments of very recent years by Breuillard, Green, Helfgott, Hrushovki, Pyber, Szabo, Tao,... are, well, very recent. And, historically this subject was (and by and large large I'd say still is) firmly 'problem solvers territory' (...) | |
| Oct 16, 2012 at 17:55 | comment | added | Salvo Tringali | Well, it seems that I've already checked them all. As for your question on possible applications of a categorical framework to approach the basic theory, I think that there is already broad evidence that looking at (fundamental, foundational) questions from a categorical point of view is likely to spread our horizons beyond any apparent imagination and open new paths towards unknown universes waiting for being explored. It may sound perhaps too romantic, but looks more than sufficient (to me) to deserve our efforts. | |
| Oct 16, 2012 at 17:39 | comment | added | Salvo Tringali | (...) of the 1996 ed. of ANT - Inverse Problems and the Geometry of Sumsets (as a side note, only in the abelian setting). (3) Grynkiewicz, in his presentation, deals uniquely with abelian groups. In the first part, he introduces the idea of a Freiman morphism of a sum-set (together with other basic material). In the second part, he discusses problems relating to what he refers to as the universal ambient group (shortly, UAG) of a sum-set. (4) Idem for Konyagin and Lev, who develop a linear algebra method (Theorem 4) to solve a couple of questions raised by Freiman himself. | |
| Oct 16, 2012 at 17:16 | comment | added | Salvo Tringali | Thanks for the references, I will check them and come back with more comments. For the moment, some preliminary considerations: (1) Additive Combinatorics introduces Freiman morphisms in Sect. 5.3 (at the least, in the 2006 ed. on my shelf), but it is strictly focused on the case of groups (as you suspected). There is just a minor remark pointing out an analogy with the differential geometry of manifolds, that's all. I may be wrong, but I guess that the 2010 ed. has not been extended in this respect. (2) It goes the same with Nathanson, who presents Freiman morphisms in Sect. 8.2 (...) | |
| Oct 16, 2012 at 16:26 | history | answered | user9072 | CC BY-SA 3.0 |