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

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 @m"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.

[1]  Hamed Hatami,et al.  Decision Trees and Influences of Variables Over Product Probability Spaces , 2006, Combinatorics, Probability and Computing.

[2]  Hamed Hatami,et al.  A remark on Bourgain's distributional inequality on the Fourier spectrum of Boolean functions , 2006 .

[3]  E. Gruner,et al.  [PCP]. , 1980, Harefuah.

[4]  N. Linial,et al.  The influence of variables in product spaces , 1992 .

[5]  J. Bourgain.  On the distribution of the fourier spectrum of Boolean functions , 2002 .

[6]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[7]  Ehud Friedgut,et al.  Influences in Product Spaces: KKL and BKKKL Revisited , 2004, Combinatorics, Probability and Computing.

[8]  K. Oleszkiewicz.  On a Nonsymmetric Version of the Khinchine-Kahane Inequality , 2003 .

[9]  Nathan Keller,et al.  On the Influences of Variables on Boolean Functions in Product Spaces , 2009, Combinatorics, Probability and Computing.

[10]  Ehud Friedgut,et al.  Boolean Functions With Low Average Sensitivity Depend On Few Coordinates , 1998, Comb..

[11]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[12]  R. O'Donnell,et al.  On the Fourier tails of bounded functions over the discrete cube , 2007 .

[13]  M. Talagrand.  On Russo's Approximate Zero-One Law , 1994 .

[14]  S. Safra,et al.  On the hardness of approximating minimum vertex cover , 2005 .

[15]  G. Kalai,et al.  Threshold Phenomena and Influence , 2005 .

[16]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[17]  Guy Kindler,et al.  Quantitative relation between noise sensitivity and influences , 2010, Comb..

[18]  M. Talagrand.  Isoperimetry, logarithmic sobolev inequalities on the discrete cube, and margulis' graph connectivity theorem , 1993 .

[19]  Michel Talagrand,et al.  On boundaries and influences , 1997, Comb..

[20]  Béla Bollobás,et al.  The critical probability for random Voronoi percolation in the plane is 1/2 , 2006 .

[21]  Michel Talagrand,et al.  How much are increasing sets positively correlated? , 1996, Comb..

[22]  P. Diaconis,et al.  LOGARITHMIC SOBOLEV INEQUALITIES FOR FINITE MARKOV CHAINS , 1996 .

[23]  Geoffrey Grimmett.  Probability on Graphs: Random graphs , 2010 .

[24]  A. Bonami.  Étude des coefficients de Fourier des fonctions de $L^p(G)$ , 1970 .