保字母提升与对数秩猜想
Alphabet-Preserving Lifting for the Log-Rank Conjecture
AI总结：
该研究针对对数秩猜想，将GPW构造的通信复杂度下界改进，通过保字母提升技术结合多色模拟定理完成，消除了原下界中的一个双对数因子。
AI中文摘要：
对于布尔通信矩阵$M$，设$D(M)$为其确定性通信复杂度，$r(M):={\rm rank}_{\real}(M)$。对数秩猜想询问$D(M)$是否为$\text{log}\thinspace r(M)$的多项式函数。目前已知的最优通用上界由Sudakov和Tomon在2025年给出，为$D(M)=O(\text{sqrt}(r(M)))$。下界方面，Göös、Pitassi和Watson在2018年构造了显式矩阵，满足$D(M)=\text{Ω}((\text{log}\thinspace r(M))^2/(\text{log}\text{log}\thinspace r(M))^2)$。我们将该下界改进为$D(M)=\text{Ω}((\text{log}\thinspace r(M))^2/\text{log}\text{log}\thinspace r(M))$。我们的构造重新考察了他们在原始非布尔字母表上的指针函数，并通过Roughgarden和Weinstein在2016年提出的多色模拟定理，用字母表值索引小工具对其进行提升。与定量显式的GPW界相比，保字母提升消除了一个$\text{log}\text{log}\thinspace r$因子。我们还针对构造所需的参数范围，给出了多色模拟定理的自包含证明。
英文摘要：
For a Boolean communication matrix $M$, let $D(M)$ denote its deterministic communication complexity and let $r(M):={\mathrm{rank}}_{\mathbb{R}}(M)$. The log-rank conjecture asks whether $D(M)$ is polynomial in $\log r(M)$. The best known general upper bound, due to Sudakov and Tomon'25, is $D(M)=O(\sqrt{r(M)})$. On the lower-bound side, G{ö}{ö}s, Pitassi, and Watson'18 constructed explicit matrices satisfying $D(M)=Ω((\log r(M))^2/(\log\log r(M))^2)$. We improve the lower bound to $D(M)=Ω((\log r(M))^2/\log\log r(M))$. Our construction revisits their pointer function over its original non-Boolean alphabet and lifts it with an alphabet-valued Index gadget, via the multicolor simulation theorem stated by Roughgarden and Weinstein'16. Compared with the quantitatively explicit GPW bound, the alphabet-preserving lift removes one factor of $\log\log r$. We also give a self-contained proof of the multicolor simulation theorem in the parameter regime required by the construction.
