We give improved separations for the query complexity analogue of the log-approximate-rank conjecture i.e. we show that there are a plethora of total Boolean functions on $n$ input bits, each of which has approximate Fourier sparsity at most $O(n^3)$ and randomized parity decision tree complexity $\Theta(n)$. This improves upon the recent work of Chattopadhyay, Mande and Sherif (JACM '20) both qualitatively (in terms of designing a large number of examples) and quantitatively (improving the gap from quartic to cubic). We leave open the problem of proving a randomized communication complexity lower bound for XOR compositions of our examples. A linear lower bound would lead to new and improved refutations of the log-approximate-rank conjecture. Moreover, if any of these compositions had even a sub-linear cost randomized communication protocol, it would demonstrate that randomized parity decision tree complexity does not lift to randomized communication complexity in general (with the XOR gadget).

中文翻译：

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对对数近似秩猜想的更强反例

我们对 log-approximate-rank 猜想的查询复杂度模拟给出了改进的分离，即我们表明在 $n$ 输入位上有大量的总布尔函数，每个都有近似的傅立叶稀疏度至多 $O(n^ 3)$和随机奇偶决策树复杂度$\Theta(n)$。这改进了 Chattopadhyay、Mande 和 Sherif (JACM '20) 最近在定性（在设计大量示例方面）和定量（改善从四次方到三次方的差距）的工作。我们留下了证明我们示例的 XOR 组合的随机通信复杂度下限的问题。线性下界将导致对对数近似秩猜想的新的和改进的反驳。此外，如果这些组合中的任何一个甚至具有亚线性成本随机通信协议，