A Quantitative Version of the Non-Abelian Idempotent Theorem

Sanders, Tom

doi:10.1007/s00039-010-0107-2

Cited by 16 publications

(16 citation statements)

References 41 publications

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“…Nevertheless, it was shown in [2] that for G = Z/N Z one can find a regular Bohr set by slightly decreasing the parameter δ. We show the same for general groups, repeating the arguments from [19,Lemma 4.25] (see also [14,Lemma 9.3]).…”

Section: Appendixsupporting

confidence: 73%

“…By the left/right invariance of • one can easily show (or consult [15,Lemma 4.1]) the normality of Bohr sets, i.e., the identity x Bohr(Γ, δ)x −1 = Bohr(Γ, δ), which holds for any x ∈ G. If Γ = {ρ}, then we simply write Bohr(ρ, δ) for Bohr(Γ, δ) (a lower bound for the size of Bohr(ρ, δ) can be found in [14,Lemma 17.3]). Further properties of Bohr sets are contained in the Appendix.…”

Section: The General Casementioning

confidence: 99%

“…Here we collect further natural properties of Bohr sets and related notions which have wellknown abelian analogs. We do this for the convenience of the reader interested in this particular form of Bohr sets; most of these results are more or less contained in [2,14,15].…”

Section: Appendixmentioning

confidence: 99%

“…Having a lower bound for the size of one-dimensional Bohr sets (see [14,Lemma 17.3] or the proposition above), one can obtain a lower bound for the size of Bohr sets with an arbitrary Γ.…”

Section: Appendixmentioning

confidence: 99%

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On the Spectral Gap and the Diameter of Cayley Graphs

Shkredov

2021

Proc. Steklov Inst. Math.

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Section: Appendixsupporting

confidence: 73%

Section: The General Casementioning

confidence: 99%

Section: Appendixmentioning

confidence: 99%

“…Having a lower bound for the size of one-dimensional Bohr sets (see [14,Lemma 17.3] or the proposition above), one can obtain a lower bound for the size of Bohr sets with an arbitrary Γ.…”

Section: Appendixmentioning

confidence: 99%

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On the Spectral Gap and the Diameter of Cayley Graphs

Shkredov

2021

Proc. Steklov Inst. Math.

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“…In the classical setting of Fourier series, functions in the Wiener algebra have absolutely convergent Fourier series, and in particular are necessarily continuous. A deep result of Sanders [13] asserts, roughly speaking, that in more general non-abelian groups G, functions in the Wiener algebra A(G) can be uniformly approximated by continuous functions "outside of a set of negligible measure". A precise version of this statement is as follows:…”

Section: An Argument Of Sandersmentioning

confidence: 99%

Noncommutative sets of small doubling

Tao

2013

European Journal of Combinatorics

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A corollary of Kneser's theorem, one sees that any finite nonempty subset A of an abelian group G = (G, +) with |A + A| ≤ (2 − ε)|A| can be covered by at most 2 ε −1 translates of a finite group H of cardinality at most (2 − ε)|A|. Using some arguments of Hamidoune, we establish an analogue in the noncommutative setting. Namely, if A is a finite non-empty subset of a nonabelian group G = (G, ·) such that |A · A| ≤ (2 − ε)|A|, then A is either contained in a right-coset of a finite group H of cardinality at most 2 ε |A|, or can be covered by at most 2 ε − 1 right-cosets of a finite group H of cardinality at most |A|. We also note some connections with some recent work of Sanders and of Petridis.

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The structure of approximate groups

Breuillard

Green

Tao³

2012

Publ.math.IHES

167

267

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Let K 1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A · A is covered by K left translates of A.The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups.We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups.To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert's fifth problem in the 1950s.Applications of our main theorem include a finitary refinement of Gromov's theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.Résumé. Soit K 1 un paramètre. On appelle groupe K-approximatif toute partie finie A d'un groupe (ou d'un groupe local) qui est symétrique, contient l'identité et est telle que A · A peutêtre recouvert par au plus K translatésà gauche de A.Le résultat principal de cet article montre que tout groupe approximatif est, grossièrement, finipar-nilpotent, ce qui répond par l'affirmativeà une conjecture de H.Helfgott et de E.Lindenstrauss.On peut interpréter ce théorème comme une généralisationà un groupe quelconque du théorème de Freiman-Ruzsa sur la structure des parties finiesà petit doublement du groupe additif des entiers relatifs.Nous commençons parétablir un principe de correspondence entre les groupes approximatifs et les groupes (ou groupes locaux) localement compacts, puis nous en déduisons de nombreuses conséquences issues d'un important article récent de Hrushovski. En particulier, nous montrons que tout groupe approximatif peut-être représenté par un groupe de Lie.Pour démontrer notre théorème principal, nous appliquons des arguments, en substance dusà Gleason pour la plupart, qui ont vus le jour dans le contexte de la solution du cinquième problème de Hilbert dans les années 50.En guise d'application du théorème principal, nous montrons une version fine du théorème de Gromov, ainsi qu'un lemme de Margulis généralisé conjecturé par Gromov, et un résultat sur la presque nilpotence des groupes fondamentaux des variétésà courbure de Ricci presque positive.

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A Quantitative Version of the Non-Abelian Idempotent Theorem

Abstract: Abstract. Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L 2 (G) to L 2

Cited by 16 publications

References 41 publications

On the Spectral Gap and the Diameter of Cayley Graphs

On the Spectral Gap and the Diameter of Cayley Graphs

Noncommutative sets of small doubling

The structure of approximate groups

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