A Freĭman-type theorem for locally compact abelian groups

Sanders, Tom

doi:10.5802/aif.2465

Cited by 2 publications

(4 citation statements)

References 10 publications

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“…However, this kind of continuous analogue is different in nature from the one we propose above, since it focuses on algebraic rather than convex structure. Another notable continuous analogue of Freiman's theorem is developed in the more general context of locally compact, abelian groups by T. Sanders [46].…”

Section: Submodularity Of Entropy and Implicationsmentioning

confidence: 99%

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

Bobkov

Madiman

2012

Journal of Functional Analysis

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We develop a reverse entropy power inequality for convex measures, which may be seen as an affine-geometric inverse of the entropy power inequality of Shannon and Stam. The specialization of this inequality to log-concave measures may be seen as a version of Milman's reverse Brunn-Minkowski inequality. The proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies, and on a study of effective supports of convex measures, both of which are of independent interest, as well as on Milman's deep technology of M -ellipsoids and on certain information-theoretic inequalities. As a by-product, we also give a continuous analogue of some Plünnecke-Ruzsa inequalities from additive combinatorics. * S. Bobkov is with the Here we recall basic definitions and the characterization of the so-called convex measures.Given −∞ ≤ κ ≤ 1, a probability measure µ on R n is called κ-concave, if it satisfies the Brunn-Minkowski-type inequalityfor all t ∈ (0, 1) and for all Borel measurable sets A, B ⊂ R n with positive measure. When κ = 0, (2.1) describes the class of log-concave measures which thus satisfy µ tA + (1 − t)B ≥ µ(A) t µ(B) 1−t .

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Section: Submodularity Of Entropy and Implicationsmentioning

confidence: 99%

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

Bobkov

Madiman

2012

Journal of Functional Analysis

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“…This result is established in [San09] where it is also noted that the relative polynomial growth hypothesis is qualitatively implied by a small doubling hypothesis (c.f. Proposition A.3 of the appendix).…”

Section: Introductionmentioning

confidence: 64%

“…Then A is contained in an Opd log 3 2dq-dimensional ball B, of positive radius, in a translation invariant pseudo-metric and |B| ď exppOpd log 2dqq|A|. This result is established in [San09] where it is also noted that the relative polynomial growth hypothesis is qualitatively implied by a small doubling hypothesis (c.f. Proposition A.3 of the appendix).…”

Section: Introductionmentioning

confidence: 66%

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Approximate groups and doubling metrics

Sanders¹

2011

Math. Proc. Camb. Phil. Soc.

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We develop a version of Freiman's theorem for a class of non-abelian groups, which includes finite nilpotent, supersolvable and solvable A-groups. To do this we have to replace the small doubling hypothesis with a stronger relative polynomial growth hypothesis akin to that in Gromov's theorem (although with an effective range), and the structures we find are balls in (left and right) translation invariant pseudo-metrics with certain well behaved growth estimates. Our work complements three other recent approaches to developing non-abelian versions of Freiman's theorem by Breuillard and Green, Fischer, Katz and Peng, and Tao.Comment: 21 pp. Corrected typos. Changed title from `From polynomial growth to metric balls in monomial groups

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A Freĭman-type theorem for locally compact abelian groups

Cited by 2 publications

References 10 publications

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

Approximate groups and doubling metrics

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