We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.
@InProceedings{kelman_et_al:LIPIcs.ITCS.2021.26, author = {Kelman, Esty and Khot, Subhash and Kindler, Guy and Minzer, Dor and Safra, Muli}, title = {{Theorems of KKL, Friedgut, and Talagrand via Random Restrictions and Log-Sobolev Inequality}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {26:1--26:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.26}, URN = {urn:nbn:de:0030-drops-135657}, doi = {10.4230/LIPIcs.ITCS.2021.26}, annote = {Keywords: Fourier Analysis, Hypercontractivity, Log-Sobolev Inequality} }
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