The log-rank conjecture, a longstanding problem in communication complexity, has persistently eluded resolution for decades. Consequently, some recent efforts have focused on potential approaches for establishing the conjecture in the special case of XOR functions, where the communication matrix is lifted from a boolean function, and the rank of the matrix equals the Fourier sparsity of the function, which is the number of its nonzero Fourier coefficients. In this note, we refute two conjectures. The first has origins in Montanaro and Osborne (arXiv'09) and is considered in Tsang et al. (FOCS'13), and the second one is due to Mande and Sanyal (FSTTCS'20). These conjectures were proposed in order to improve the best-known bound of Lovett (STOC'14) regarding the log-rank conjecture in the special case of XOR functions. Both conjectures speculate that the set of nonzero Fourier coefficients of the boolean function has some strong additive structure. We refute these conjectures by constructing two specific boolean functions tailored to each.

翻译：log-rank猜想是通信复杂度领域一个长期未决的问题，几十年来始终未能得到解决。因此，近年来的研究聚焦于探索在XOR函数这一特殊情形下证明该猜想的潜在方法。在XOR函数中，通信矩阵由布尔函数提升而成，矩阵的秩等于该函数的傅里叶稀疏度，即其非零傅里叶系数的个数。本文反驳了两个猜想：第一个猜想源于Montanaro和Osborne（arXiv'09），并在Tsang等人（FOCS'13）的研究中有所涉及；第二个猜想由Mande和Sanyal（FSTTCS'20）提出。这些猜想旨在改进Lovett（STOC'14）就XOR函数特例所给出的log-rank猜想最佳已知上界。两个猜想均假设布尔函数的非零傅里叶系数集合具有某种强加法结构。我们通过为每个猜想分别构造特定的布尔函数，成功反驳了这两个猜想。