Abstract
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. Our proof works in the case where the expectation of the function is bounded away from 0 and 1. In contrast, [6] deals with Boolean functions of expectation O(1/ n) whose Fourier transform is highly concentrated on the first two irreducible representations of Sn. These need not be close to dictatorships; rather, they must be close to a union of a constant number of cosets of point-stabilizers.
Original language | English |
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Pages (from-to) | 494-530 |
Number of pages | 37 |
Journal | Random Structures and Algorithms |
Volume | 46 |
Issue number | 3 |
DOIs | |
State | Published - 1 May 2015 |
Externally published | Yes |
Keywords
- Fourier transform
- Stability
- Symmetric group
ASJC Scopus subject areas
- Software
- General Mathematics
- Computer Graphics and Computer-Aided Design
- Applied Mathematics