The level-$k$ $\ell_1$-Fourier weight of a Boolean function refers to the sum of absolute values of its level-$k$ Fourier coefficients. Fourier growth refers to the growth of these weights as $k$ grows. It has been extensively studied for various computational models, and bounds on the Fourier growth, even for the first few levels, have proven useful in learning theory, circuit lower bounds, pseudorandomness, and quantum-classical separations. We investigate the Fourier growth of certain functions that naturally arise from communication protocols for XOR functions (partial functions evaluated on the bitwise XOR of the inputs to Alice and Bob). If a protocol $\mathcal C$ computes an XOR function, then $\mathcal C(x,y)$ is a function of the parity $x\oplus y$. This motivates us to analyze the XOR-fiber of $\mathcal C$, defined as $h(z):=\mathbb E_{x,y}[\mathcal C(x,y)|x\oplus y=z]$. We present improved Fourier growth bounds for the XOR-fibers of protocols that communicate $d$ bits. For the first level, we show a tight $O(\sqrt d)$ bound and obtain a new coin theorem, as well as an alternative proof for the tight randomized communication lower bound for Gap-Hamming. For the second level, we show an $d^{3/2}\cdot\mathrm{polylog}(n)$ bound, which improves the previous $O(d^2)$ bound by Girish, Raz, and Tal (ITCS 2021) and implies a polynomial improvement on the randomized communication lower bound for the XOR-lift of Forrelation, extending its quantum-classical gap. Our analysis is based on a new way of adaptively partitioning a relatively large set in Gaussian space to control its moments in all directions. We achieve this via martingale arguments and allowing protocols to transmit real values. We also show a connection between Fourier growth and lifting theorems with constant-sized gadgets as a potential approach to prove optimal bounds for the second level and beyond.

翻译：摘要：布尔函数的第$k$层$\ell_1$-傅里叶权重是指其第$k$层傅里叶系数绝对值之和。傅里叶增长指的是这些权重随$k$增长的变化趋势。该性质已在多种计算模型中受到广泛研究，即使仅针对前几层的傅里叶增长边界，也在学习理论、电路下界、伪随机性以及量子-经典分离等领域展现出重要价值。我们研究了由XOR函数通信协议（基于Alice和Bob输入的按位异或计算的部分函数）自然产生的一类函数的傅里叶增长。若协议$\mathcal C$计算一个XOR函数，则$\mathcal C(x,y)$是奇偶性$x\oplus y$的函数。这促使我们分析$\mathcal C$的XOR纤维，定义为$h(z):=\mathbb E_{x,y}[\mathcal C(x,y)|x\oplus y=z]$。对于通信$d$比特的协议，我们提出了其XOR纤维的改进傅里叶增长边界。针对第一层，我们证明了紧致的$O(\sqrt d)$边界，并由此得到一个新的硬币定理，以及Gap-Hamming问题紧致随机通信下界的替代证明。针对第二层，我们证明了$d^{3/2}\cdot\mathrm{polylog}(n)$边界，改进了Girish、Raz和Tal (ITCS 2021) 此前得到的$O(d^2)$边界，并蕴含了Forrelation的XOR提升随机通信下界的多项式改进，从而扩展了其量子-经典差距。我们的分析基于一种新方法：在高斯空间中对一个相对较大的集合进行自适应划分，以控制其在所有方向上的矩。我们通过鞅论证并允许协议传输实数值来实现这一目标。我们还展示了傅里叶增长与基于常数大小子件的提升定理之间的联系，这为证明第二层及更高层的最优边界提供了潜在途径。