小灵敏度集的大小：Simon引理的推广
Size of Sets with Small Sensitivity: a Generalization of Simon's Lemma
摘要：
研究了小灵敏度集$S\subseteq\{0, 1\}^n$的结构。著名的西蒙引理说，任何灵敏度为$s$的$S\subseteq\{0, 1\}^n$的大小必须至少为$2^{n-s}$。这一结果对于证明布尔函数的灵敏度下界是有用的，并应用于并行计算理论和“灵敏度与块灵敏度”猜想。在本文中，我们将深入研究这些集合的大小及其结构。我们发现了一个意想不到的“缺口定理”：如果$S\subseteq\{0, 1\}^n$具有$s$的敏感性，那么我们要么有$|S|=2^{n-s}$，要么有$|S|\geq \frac{3}{2} 2^{n-s}$。这是通过将这些集合分类为可以从$n'<n$的$\{0, 1\}^{n'}$的低灵敏度子集构造的集合和不能以这种方式构造的不可约集合，然后证明不可约集合的大小的下限。这为布尔超立方体$\{0, 1\}^n$的低灵敏度子集的结构提供了新的见解。
英文摘要：
We study the structure of sets $S\subseteq\{0, 1\}^n$ with small sensitivity. The well-known Simon's lemma says that any $S\subseteq\{0, 1\}^n$ of sensitivity $s$ must be of size at least $2^{n-s}$. This result has been useful for proving lower bounds on sensitivity of Boolean functions, with applications to the theory of parallel computing and the "sensitivity vs. block sensitivity" conjecture. In this paper, we take a deeper look at the size of such sets and their structure. We show an unexpected "gap theorem": if $S\subseteq\{0, 1\}^n$ has sensitivity $s$, then we either have $|S|=2^{n-s}$ or $|S|\geq \frac{3}{2} 2^{n-s}$. This is shown via classifying such sets into sets that can be constructed from low-sensitivity subsets of $\{0, 1\}^{n'}$ for $n'<n$ and irreducible sets which cannot be constructed in such a way and then proving a lower bound on the size of irreducible sets. This provides new insights into the structure of low sensitivity subsets of the Boolean hypercube $\{0, 1\}^n$.
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