Boolean functions whose Fourier transform is concentrated on the first two levels
Advances in Applied Mathematics
Thresholds and Expectation Thresholds
Combinatorics, Probability and Computing
Collective coin flipping, robust voting schemes and minima of Banzhaf values
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
The influence of variables on Boolean functions
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
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We show that a set A ⊂ {0, 1}n with edge-boundary of size at most \[|A| (\log_{2}(2^{n}/|A|) + \epsilon)\] can be made into a subcube by at most (2ε/log2(1/ε))|A| additions and deletions, provided ε is less than an absolute constant. We deduce that if A ⊂ {0, 1}n has size 2t for some t ∈ ℕ, and cannot be made into a subcube by fewer than δ|A| additions and deletions, then its edge-boundary has size at least \[|A| \log_{2}(2^{n}/|A|) + |A| \delta \log_{2}(1/\delta) = 2^{t}(n-t+\delta \log_{2}(1/\delta)),\] provided δ is less than an absolute constant. This is sharp whenever δ = 1/2j for some j ∈ {1, 2,. . ., t}.