Direct Sum Theorems From Fortification

We revisit the direct sum theorems in communication complexity which askes whether the resource to solve $n$ communication problems together is (approximately) the sum of resources to solve these problems separately. Our work starts with the observation that Meir and Dinur's fortification lemma for protocol size over rectangles 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. Given a communication problem $S\subseteq (X\times Y) \times Z$, let $\Lambda \subseteq X\times Y$ be a $\delta$-fooling set of $S$, then given any subset $\tilde{\Lambda} \subseteq \Lambda$ such that $|\tilde{\Lambda}|/{|\Lambda|} > \delta$, there is no monochromatic rectangle that covers the subset $\tilde{\Lambda}$. Particularly, there is a $\frac{16\log|X| |Y|}{\mathsf{ Cov}(S)}$-fooling set of communication problem $S$. With this fact, we are able to reprove the classic direct sum theorem of cover number with a simple double counting argument. And 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 $\mathsf{L}$ denote the protocol szie, 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$.We also prove a tight cover number lower bound for the agree problem introduced by Amos Beimel et al.