Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-25T04:54:26.935Z Has data issue: false hasContentIssue false
  • English
  • Français

FKN theorem for the multislice, with applications

Part of: Circuits, networks Inequalities Harmonic analysis in several variables

Published online by Cambridge University Press:  18 October 2019

Yuval Filmus
Yuval Filmus*
Affiliation:
Computer Science Department, Technion – Israel Institute of Technology
*
Email: yuvalfi@cs.technion.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

The Friedgut–Kalai–Naor (FKN) theorem states that if ƒ is a Boolean function on the Boolean cube which is close to degree one, then ƒ is close to a dictator, a function depending on a single coordinate. The author has extended the theorem to the slice, the subset of the Boolean cube consisting of all vectors with fixed Hamming weight. We extend the theorem further, to the multislice, a multicoloured version of the slice.

As an application, we prove a stability version of the edge-isoperimetric inequality for settings of parameters in which the optimal set is a dictator.

MSC classification

Primary: 26D07: Inequalities involving other types of functions
Secondary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 94C10: Switching theory, application of Boolean algebra; Boolean functions
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 29 , Issue 2 , March 2020 , pp. 200 - 212
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Taub Fellow, supported by the Taub Foundations. The research was funded by ISF grant 1337/16.

References

Alon, N., Dinur, I., Friedgut, E. and Sudakov, B. (2004) Graph products, Fourier analysis and spectral techniques. Geom. Funct. Anal. 14 913–940.CrossRefGoogle Scholar
Diaconis, P. and Shahshahani, M. (1981) Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.CrossRefGoogle Scholar
Dikstein, Y., Dinur, I., Filmus, Y. and Harsha, P. (2018) Boolean function analysis on high-dimensional expanders. In Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques (APPROX/RANDOM 2018), Vol. 116 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum für Informatik, pp. 38:1–38:20.Google Scholar
Ellis, D., Filmus, Y. and Friedgut, E. (2015) A quasi-stability result for dictatorships in Sn. Combinatorica 35 573–618.CrossRefGoogle Scholar
Ellis, D., Filmus, Y. and Friedgut, E. (2015) A stability result for balanced dictatorships in Sn. Random Struct. Alg . 46 494–530.CrossRefGoogle Scholar
Ellis, D., Friedgut, E. and Pilpel, H. (2011) Intersecting families of permutations. J. Amer. Math. Soc. 24 649–682.CrossRefGoogle Scholar
Filmus, Y. (2016) Friedgut–Kalai–Naor theorem for slices of the Boolean cube. Chic. J. Theoret. Comput. Sci. 2016 14:1–14:17.Google Scholar
Filmus, Y. and Ihringer, F. (2019) Boolean degree 1 functions on some classical association schemes. J. Combin. Theory Ser. A 162 241–270.CrossRefGoogle Scholar
Filmus, Y., O’Donnell, R. and Wu, X. (2018) A log-Sobolev inequality for the multislice, with applications. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019), Vol. 124 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl–Leibniz-Zentrum für Informatik, pp. 34:1–34:12.Google Scholar
Friedgut, E., Kalai, G. and Naor, A. (2002) Boolean functions whose Fourier transform is concentrated on the first two levels and neutral social choice. Adv. Appl. Math. 29 427–437.CrossRefGoogle Scholar
Frobenius, G. (1900) Über die Charaktere der symmetrischen Gruppe. Sitzungsberichte der Königlich preussischen Akademie der Wissenschaften zu Berlin, pp. 516–534.Google Scholar
Jendrej, J., Oleszkiewicz, K. and Wojtaszczyk, J. O. (2015) On some extensions of the FKN theorem. Theory Comput. 11 445–469.CrossRefGoogle Scholar
Nayar, P. (2014) FKN theorem on the biased cube. Colloq. Math. 137 253–261.CrossRefGoogle Scholar
O’Donnell, R. (2014) Analysis of Boolean Functions, Cambridge University Press.CrossRefGoogle Scholar
Rubinstein, A. (2012) Boolean functions whose Fourier transform is concentrated on pairwise disjoint subsets of the input. Master’s thesis, Tel Aviv University, Israel.Google Scholar
Sagan, B. E. (2001) The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, second edition, Vol. 203 of Graduate Texts in Mathematics, Springer.CrossRefGoogle Scholar