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We show that if $fcolon S_n to {0,1}$ is $epsilon$-close to linear in $L_2$ and $mathbb{E}[f] leq 1/2$ then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and moreover this is sharp: any such union is close to linear. This constitutes a sharp Friedgut-Kalai-Naor theorem for the symmetric group. Using similar techniques, we show that if $fcolon S_n to mathbb{R}$ is linear, $Pr[f otin {0,1}] leq epsilon$, and $Pr[f = 1] leq 1/2$, then $f$ is $O(epsilon)$-close to a union of mostly disjoint cosets, and this is also sharp; and that if $fcolon S_n to mathbb{R}$ is linear and $epsilon$-close to ${0,1}$ in $L_infty$ then $f$ is $O(epsilon)$-close in $L_infty$ to a union of disjoint cosets.
We show that a Boolean degree $d$ function on the slice $binom{[n]}{k} = { (x_1,ldots,x_n) in {0,1} : sum_{i=1}^n x_i = k }$ is a junta, assuming that $k,n-k$ are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube
Let $mathcal{F}$ and $mathcal{G}$ be two $t$-uniform families of subsets over $[k] = {1,2,...,k}$, where $|mathcal{F}| = |mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in $mathcal{F}$ and $mat
We study regular graphs in which the random walks starting from a positive fraction of vertices have small mixing time. We prove that any such graph is virtually an expander and has no small separator. This answers a question of Pak [SODA, 2002]. As
We show that almost all n-bit Boolean functions have bounded-error quantum query complexity at least n/2, up to lower-order terms. This improves over an earlier n/4 lower bound of Ambainis, and shows that van Dams oracle interrogation is essentially
An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an alm