A simple reduction from a biased measure on the discrete cube to the uniform measure

Nathan Keller

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arXiv · 1001.1167

A simple reduction from a biased measure on the discrete cube to the uniform measure

Nathan Keller

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Abstract

We show that certain statements related to the Fourier-Walsh expansion of functions with respect to a biased measure on the discrete cube can be deduced from the respective results for the uniform measure by a simple reduction. In particular, we present simple generalizations to the biased measure $μ_p$ of the Bonami-Beckner hypercontractive inequality, and of Talagrand's lower bound on the size of the boundary of subsets of the discrete cube. Our generalizations are tight up to constant factors.

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Nathan Keller. 2010-11-24. A simple reduction from a biased measure on the discrete cube to the uniform measure. https://arxiv.org/abs/1001.1167

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