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：对数秩猜想起源于通信复杂度领域，作为长期悬而未决的问题，数十年来始终未能得到解决。因此，近期研究重点转向探索在XOR函数这一特殊情形下证明该猜想的潜在途径。在此类函数中，通信矩阵由布尔函数提升而来，矩阵秩等于该函数的傅里叶稀疏度（即非零傅里叶系数的个数）。本文反驳了两个猜想：第一个猜想源自Montanaro与Osborne（arXiv'09），并得到Tsang等人（FOCS'13）的探讨；第二个猜想由Mande与Sanyal（FSTTCS'20）提出。这些猜想旨在改进Lovett（STOC'14）关于XOR函数特例下log-rank猜想的最优已知界。两个猜想均推测布尔函数的非零傅里叶系数集具有某种强加法结构。我们通过分别构造两个特定的布尔函数，对这两个猜想进行了反驳。