A stability result for balanced dictatorships in Sn

We prove that a balanced Boolean function on Sn whose Fourier transform is highly concentrated on the first two irreducible representations of Sn, is close in structure to a dictatorship, a function which is determined by the image or pre-image of a single element. As a corollary, we obtain a stability result concerning extremal isoperimetric sets in the Cayley graph on Sn generated by the transpositions.

[1]  Haran Pilpel,et al.  Intersecting Families of Permutations , 2010, 1011.3342.

[2]  S. Safra,et al.  Noise-Resistant Boolean-Functions are Juntas , 2003 .

[3]  A. Terras.  Fourier Analysis on Finite Groups and Applications: Index , 1999 .

[4]  Noam Nisan,et al.  On the degree of boolean functions as real polynomials , 1992, STOC '92.

[5]  E. Friedgut,et al.  LOW-DEGREE BOOLEAN FUNCTIONS ON $S_{n}$ , WITH AN APPLICATION TO ISOPERIMETRY , 2015, Forum of Mathematics, Sigma.

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

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

[8]  A. C. Berry.  The accuracy of the Gaussian approximation to the sum of independent variates , 1941 .

[9]  Bruce E. Sagan,et al.  The symmetric group - representations, combinatorial algorithms, and symmetric functions , 2001, Wadsworth & Brooks / Cole mathematics series.

[10]  Stefan Gerhold,et al.  Probability Inequalities , 2017 .

[11]  N. Alon,et al.  il , , lsoperimetric Inequalities for Graphs , and Superconcentrators , 1985 .

[12]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[13]  Yuval Filmus,et al.  A quasi-stability result for dictatorships in Sn , 2012, Comb..

[14]  Yuval Filmus,et al.  A quasi-stability result for low-degree Boolean functions on Sn , 2012 .

[15]  E. M. Elsayed,et al.  DIFFERENCE EQUATIONS , 2001 .

[16]  Ryan O'Donnell,et al.  Gaussian noise sensitivity and Fourier tails , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[17]  Yuval Filmus,et al.  Spectral Methods in Extremal Combinatorics , 2014 .

[18]  P. Diaconis,et al.  Generating a random permutation with random transpositions , 1981 .

[19]  David Ellis,et al.  Stability for t-intersecting families of permutations , 2008, J. Comb. Theory, Ser. A.

[20]  Assaf Naor,et al.  Boolean functions whose Fourier transform is concentrated on the first two levels , 2002, Adv. Appl. Math..