On the Converse of Talagrand's Influence Inequality

Saleet Klein; Amit Levi; Muli Safra; Clara Shikhelman; Yinon Spinka

On the Converse of Talagrand's Influence Inequality

Discrete Mathematics 2015-06-24 v2 Combinatorics

Authors: Saleet Klein , Amit Levi , Muli Safra , Clara Shikhelman , Yinon Spinka

Abstract

In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $0<a_1,a_2,\ldots,a_n\le 1$ such that $\sum_{j=1}^n a_j/(1-\log a_j)\ge C$ for some constant $C>0$ , it is possible to find a roughly balanced Boolean function $f$ such that $\textrm{Inf}_j[f] < a_j$ for every $1 \le j \le n$ .

Keywords

Talagrand's influence inequality KKL theorem boolean function influence converse balanced function discrete mathematics

Cite

@article{arxiv.1506.06325,
  title  = {On the Converse of Talagrand's Influence Inequality},
  author = {Saleet Klein and Amit Levi and Muli Safra and Clara Shikhelman and Yinon Spinka},
  journal= {arXiv preprint arXiv:1506.06325},
  year   = {2015}
}

On the Converse of Talagrand's Influence Inequality

Abstract

Keywords

Cite

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