FKN theorem for the multislice, with applications

FKN Theorem on the biased cube

In this note we consider Boolean functions defined on the discrete cube equipped with a biased product probability measure. We prove that if the spectrum of such a function is concentrated on the first two Fourier levels, then the function is close to a certain function of one variable. Moreover, in the symmetric case we prove that if a [-1,1]-valued function defined on the discrete cube is close to a certain affine function, then it is also close to a [-1,1]-valued affine function.

A log-Sobolev inequality for the multislice, with applications

Let $κ\in \mathbb{N}_+^\ell$ satisfy $κ_1 + \dots + κ_\ell = n$ and let $\mathcal{U}_κ$ denote the "multislice" of all strings $u$ in $[\ell]^n$ having exactly $κ_i$ coordinates equal to $i$, for all $i \in [\ell]$. Consider the Markov chain on $\mathcal{U}_κ$, where a step is a random transposition of two coordinates of $u$. We show that the log-Sobolev constant $ρ_κ$ for the chain satisfies $$(ρ_κ)^{-1} \leq n \sum_{i=1}^{\ell} \tfrac{1}{2} \log_2(4n/κ_i),$$ which is sharp up to constants whenever $\ell$ is constant. From this, we derive some consequences for small-set expansion and isoperim…

二元格拉斯曼方案的FKN定理

An FKN Theorem for the Binary Grassmann Scheme

本文将Friedgut等提出的FKN定理推广到二元格拉斯曼方案，证明了该方案中与1次函数接近的二元函数的结构性质，为相关组合分析提供了新结果。

欣钦不等式的小系数变体与精确π/2定理

A Sharp Small-Coefficient Variant of Khintchine's Inequality and the Sharp $π/2$ Theorem

本文针对有界最大系数的归一化加权拉德马赫和，证明了精确二次正态近似的误差界，结合零偏斯坦方法与小球概率估计验证了收敛速率最优性，并将其应用于改进布尔函数的FKN定理（精确π/2定理）。

论 $Φ$-稳定性及相关猜想

On the $Φ$-Stability and Related Conjectures

本文研究离散立方体上 Boolean 函数的最大 $Φ$-稳定性，利用 Fourier 分析、优化理论和改进的 FKN 定理给出上界，并部分证明三个相关猜想。

Boolean functions whose Fourier transform is concentrated on pairwise disjoint subsets of the input

We consider Boolean functions f:{-1,1}^n->{-1,1} that are close to a sum of independent functions on mutually exclusive subsets of the variables. We prove that any such function is close to just a single function on a single subset. We also consider Boolean functions f:R^n->{-1,1} that are close, with respect to any product distribution over R^n, to a sum of their variables. We prove that any such function is close to one of the variables. Both our results are independent of the number of variables, but depend on the variance of f. I.e., if f is ε*Var(f)-close to a sum of independent functions…