Comments on: 254A, Notes 7: Models of ultra approximate groups http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Tue, 21 May 2013 12:32:45 +0000 hourly 1 http://wordpress.com/ By: Lou van den Dries http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-127276 Wed, 08 Feb 2012 19:39:27 +0000 http://terrytao.wordpress.com/?p=5413#comment-127276 The “less than” sign in (2) in the proof of Sanders’ lemma should be a
“greater than” sign, I think.

[Corrected, thanks - T.]

]]> By: Lou van den Dries http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-127275 Wed, 08 Feb 2012 19:36:22 +0000 http://terrytao.wordpress.com/?p=5413#comment-127275 Some typos in the proof of Sanders’ Lemma:
. Two displays later, (1/m)|A^2| should be
(1/m)|A’A|, and the right hand sides in the next two displays should be
(1-(1/m)) |A’A| and (1-(1/m))f(t)|A|. In the subsequent two displays
the right hand side is missing a factor |A|.

[Corrected, thanks - T.]

]]> By: Terence Tao http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-110917 Tue, 29 Nov 2011 01:03:19 +0000 http://terrytao.wordpress.com/?p=5413#comment-110917 Thanks for the corrections!

]]> By: Nick Cook http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-110904 Mon, 28 Nov 2011 23:53:40 +0000 http://terrytao.wordpress.com/?p=5413#comment-110904 Some small typos: in Lemma 10 do you want \delta_1, \delta_2 as in the proof?

Exercise 29 – N should be n in the definition of A_n.

Exercise 31, part (iii) – I think should be project to one of the two factors G_0, rather than two factors of G_0.

]]> By: Nick Cook http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-110902 Mon, 28 Nov 2011 23:46:48 +0000 http://terrytao.wordpress.com/?p=5413#comment-110902 In the proof of the Normal Sanders Lemma, doesn’t application of the Rusza covering lemma give that A is covered by O_{K,m}(1) left translates of S^2, not S? Everything still works if we take S such that S^{8m} is contained in A^4 at the start – I just want to check my understanding that the Rusza covering lemma converts statements like |S|\gg_{K,m}|A| to statements about A being covered by translates of S^2 rather than S.

]]> By: Terence Tao http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-105410 Tue, 15 Nov 2011 03:44:10 +0000 http://terrytao.wordpress.com/?p=5413#comment-105410 Oops, there were several things wrong with that exercise; I’ve had to rewrite it a bit to address the issues you found. Hopefully it is now ok…

]]> By: Ben Hayes http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-105360 Tue, 15 Nov 2011 00:33:37 +0000 http://terrytao.wordpress.com/?p=5413#comment-105360 In Exercise 31 should there be a -n somewhere? I think, e.g. you may want A_{n}=\{(i,x):i\in \{-1,0,1\},x\in ({\bf Z}/2{\bf Z})^{\{-n,\cdots,n\}} I’m having difficulty showing that A contains the inverse image of an open neighborhood of the identity in G\times_{\mathbf{Z}}G.

]]> By: Ben Hayes http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-104988 Mon, 14 Nov 2011 05:53:03 +0000 http://terrytao.wordpress.com/?p=5413#comment-104988 Should the m in the displayed equation on part (iv) of Exercise 23 be a displayed -m? Otherwise this part doesn’t seem as helpful.

[Oops, corrected, thanks - T.]

]]> By: 254A, Notes 9: Applications of the structural theory of approximate groups « What’s new http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-104883 Mon, 14 Nov 2011 01:20:46 +0000 http://terrytao.wordpress.com/?p=5413#comment-104883 [...] but we will give here an alternate argument relying on a version of the Croot-Sisask lemma used in Notes 7 which is a little weaker with regards to quantitative bounds, but is slightly simpler technically [...]

]]> By: 254A, Notes 8: The microstructure of approximate groups « What’s new http://terrytao.wordpress.com/2011/10/27/254a-notes-7-models-of-ultra-approximate-groups/#comment-101198 Sun, 06 Nov 2011 17:22:45 +0000 http://terrytao.wordpress.com/?p=5413#comment-101198 [...] macroscopic structure of these objects is well described by the Hrushovski Lie model theorem from the previous set of notes, which informally asserts that the macroscopic structure of an (ultra) approximate group can be [...]

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