发表机构
Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Friedgut猜想：任意函数存在大小为O(n/√log n)的联盟，能以高概率控制输出，该界与字母表大小无关，并应用于集体抛硬币，给出首个次线性坏玩家数界。
AI 中文摘要
我们证明，对于每个常数 $\varepsilon>0$ 和每个函数 $f:\Sigma^n\to\{0, 1\}$，存在一个由 $O(n/\sqrt{\log n})$ 个坐标组成的联盟和一个目标输出 $b\in\{0, 1\}$，使得在剩余坐标被均匀且独立地采样后，该联盟可以选择其值，以至少 $1-\varepsilon$ 的概率使输出等于 $b$。该界与字母表大小无关，并且也适用于 $[0,1]^n$ 上的单调布尔函数，从而解决了 Friedgut 的一个猜想（Combinatorics, Probability and Computing, 2004）。与布尔立方体情形不同——在布尔立方体中，Kahn、Kalai 和 Linial（FOCS, 1988）给出了 $O(n/\log n)$ 的联盟界——此前并不知道与字母表大小无关的次线性界。在集体抛硬币中，我们的结果给出了首个次线性界，即在任何具有独立均匀消息的单轮协议中，无论消息长度如何，迫使固定输出以至少 $1-\varepsilon$ 概率出现所需的坏玩家数量。我们证明中的一个关键要素是一种编码，它使我们能够将函数在乘积空间上的影响力与编码函数的 $p$-偏置影响力联系起来。然后，我们依赖 Hatami（Annals of Mathematics, 2012）关于具有小 $p$-偏置影响力的函数的结构定理来偏置编码函数。
英文摘要
We prove that, for every constant $\varepsilon>0$ and every function $f:Σ^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ with probability at least $1-\varepsilon$. The bound is independent of the alphabet size and also holds for monotone Boolean functions on $[0,1]^n$, resolving a conjecture of Friedgut (Combinatorics, Probability and Computing, 2004). Unlike the Boolean cube setting, where Kahn, Kalai, and Linial (FOCS, 1988) give a coalition bound of $O(n/\log n)$, no sublinear bound independent of the alphabet size was previously known. In collective coin flipping, our result gives the first sublinear bound on the number of bad players needed to force a fixed output with probability at least $1-\varepsilon$ in any one-round protocol with independent uniform messages, regardless of the message length. A key ingredient in our proof is an encoding that lets us relate the influence of a function on a product space to the $p$-biased influence of the encoded function. We then rely on a structure theorem of Hatami (Annals of Mathematics, 2012) for functions with small $p$-biased influence to bias the encoded function.
