We revisit the direct sum questions in communication complexity which asks whether the resource needed to solve $n$ communication problems together is (approximately) the sum of resources needed to solve these problems separately. Our work starts with the observation that Dinur and Meir's fortification lemma can be generalized to a general fortification lemma for a sub-additive measure over set. By applying this lemma to the case of cover number, we obtain a dual form of cover number, called "$\delta$-fooling set" which is a generalized fooling set. Any rectangle which contains enough number of elements from a $\delta$-fooling set can not be monochromatic. With this fact, we are able to reprove the classic direct sum theorem of cover number with a simple double counting argument. Formally, let $S \subseteq (A\times B) \times O$ and $T \subseteq (P\times Q) \times Z$ be two communication problems, $ \log \mathsf{Cov}\left(S\times T\right) \geq \log \mathsf{Cov}\left(S\right) + \log\mathsf{Cov}(T) -\log\log|P||Q|-4.$ where $\mathsf{Cov}$ denotes the cover number. One issue of current deterministic direct sum theorems about communication complexity is that they provide no information when $n$ is small, especially when $n=2$. In this work, we prove a new direct sum theorem about protocol size which imply a better direct sum theorem for two functions in terms of protocol size. Formally, let $\mathsf{L}$ denotes complexity of the protocol size of a communication problem, given a communication problem $F:A \times B \rightarrow \{0,1\}$, $ \log\mathsf{L}\left(F\times F\right)\geq \log \mathsf{L}\left(F\right) +\Omega\left(\sqrt{\log\mathsf{L}\left(F\right)}\right)-\log\log|A||B| -4$. All our results are obtained in a similar way using the $\delta$-fooling set to construct a hardcore for the direct sum problem.

翻译：我们重新审视通信复杂性中的直接和问题，该问题探究解决$n$个通信问题所需的总资源是否（近似地）等于分别解决这些问题的资源之和。本研究始于一个观察：Dinur和Meir的加固引理可推广为针对集合上子可加测度的一般性加固引理。将该引理应用于覆盖数情形时，我们得到了覆盖数的对偶形式——称为"$\delta$-欺骗集"，这是一种广义的欺骗集。任何包含足够多$\delta$-欺骗集元素的矩形都不可能是单色的。基于这一事实，我们通过简单的双计数论证重新证明了经典的覆盖数直接和定理。形式化地，设$S \subseteq (A\times B) \times O$和$T \subseteq (P\times Q) \times Z$为两个通信问题，则有$\log \mathsf{Cov}\left(S\times T\right) \geq \log \mathsf{Cov}\left(S\right) + \log\mathsf{Cov}(T) -\log\log|P||Q|-4$，其中$\mathsf{Cov}$表示覆盖数。当前关于通信复杂性的确定性直接和定理存在一个问题：当$n$较小时（尤其是$n=2$的情况），它们无法提供有效信息。在本工作中，我们证明了关于协议规模的新直接和定理，该定理在协议规模维度上为两个函数的情形提供了更优的直接和结论。形式化地，设$\mathsf{L}$表示通信问题的协议规模复杂度，对于通信问题$F:A \times B \rightarrow \{0,1\}$，有$\log\mathsf{L}\left(F\times F\right)\geq \log \mathsf{L}\left(F\right) +\Omega\left(\sqrt{\log\mathsf{L}\left(F\right)}\right)-\log\log|A||B| -4$。我们所有结果均采用类似方法获得——通过构造$\delta$-欺骗集来为直接和问题建立核心困难。