A structure theorem for Boolean functions with small total influences

Hatami, Hamed

doi:10.4007/annals.2012.176.1.9

Cited by 34 publications

(38 citation statements)

References 18 publications

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“…In contrast to the results of [4,12,19], which describe the structure of families with total influence within a constant factor of the minimum, our Theorem 1.5 describes the structure of Boolean functions with total influence 'very' close to the minimum. On the other hand, the structure we obtain is very strong -namely, closeness to a genuinely extremal family.…”

Section: Related Workcontrasting

confidence: 67%

On the structure of subsets of the discrete cube with small edge boundary

Ellis¹,

Keller

Lifshitz

2018

Discrete Anal

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Abstract:The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers m and n, the minimum size g n (m) of the edge boundary of an m-element subset of {0, 1} n ; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on {0, 1} n . We show that for any m-element subset F ⊂ {0, 1} n and any integer l, if the edge boundary of F has size at most g n (m) + l, then there exists an extremal family G ⊂ {0, 1} n such that |F∆G| ≤ Cl, where C is an absolute constant. This is best possible, up to the value of C. Our result can be seen as a 'stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli [15] for the isoperimetric inequality in Euclidean space.

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Section: Related Workcontrasting

confidence: 67%

On the structure of subsets of the discrete cube with small edge boundary

Ellis¹,

Keller

Lifshitz

2018

Discrete Anal

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“…The case εµ < ∆ ′ µ ′ is similar, so we shall skip some of the details. Note that 3kp > ε, by (27), and hence pn/k < (3/ε)p 2 n = (3C/ε) log n. Observe also that (30)…”

Section: The Group Z 2nmentioning

confidence: 84%

Random sum-free subsets of abelian groups

2013

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We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group G. In particular, we determine the threshold above which, with high probability as |G| → ∞, each such subset is contained in some maximum-size sum-free subset of G, whenever q divides |G| for some (fixed) prime q with q ≡ 2 (mod 3). Moreover, in the special case G = Z 2n , we determine the sharp threshold for the above property. The proof uses recent 'transference' theorems of Conlon and Gowers, together with stability theorems for sum-free subsets of Abelian groups.

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“…(2) For the case where K is bounded, µ p ( f ) is bounded away from zero and one, but log(1/ p)/ log n is bounded away from zero, there are important theorems by Friedgut [1999] and Bourgain [1999] (see below) and Hatami [2012]. These results have important applications for proving sharp threshold theorems.…”

Section: Isoperimetric Inequalities and Russo's 0-1 Lawmentioning

confidence: 99%

Around two theorems and a lemma by Lucio Russo

Benjamini

Kalai

2018

Math. Mech. Compl. Sys.

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We describe two directions of study following early work of Lucio Russo. The first direction follows the famous Russo-Seymour-Welsh (RSW) theorem. We describe an RSW-type conjecture by the first author which, if true, would imply a coarse version of conformal invariance for critical planar percolation. The second direction is the study of "Russo's lemma" and "Russo's 0-1 law" for threshold behavior of Boolean functions. We mention results by Friedgut, Bourgain, and Hatami, and present a conjecture by Jeff Kahn and the second author, which may allow applications for finding critical probabilities.

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A structure theorem for Boolean functions with small total influences

Cited by 34 publications

References 18 publications

On the structure of subsets of the discrete cube with small edge boundary

On the structure of subsets of the discrete cube with small edge boundary

Random sum-free subsets of abelian groups

Around two theorems and a lemma by Lucio Russo

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