Parity Decision Trees of Depth $O(4^d \log\|\widehat f\|_1)$ for Boolean Functions $f$ of $F_2$-Degree $d$

Abstract

Let $f\colon F_2^n\to\{-1,+1\}$ have $F_2$-degree $d\ge2$. We prove that its deterministic parity decision tree depth satisfies \[ D_{\oplus} (f)= O(4^d \log_2\|\widehat f\|_1) = O(4^d \log_2 s(f)), \] where $\|\widehat f\|_1=\sum_ξ|\widehat f(ξ)|$ is the Fourier spectral norm and $s(f)$ is the Fourier sparsity. The bound is linear in the logarithm of the spectral norm for every fixed degree, with singly exponential dependence on $d$. This improves the previously best known bound of $O(2^{d^2/2} \log^{d-2} \|\widehat f\|_1)$ due to Tsang, Wong, Xie and Zhang (FOCS 2013). Our result establishes the log-rank conjecture for $\mathrm{XOR}$ functions of $F_2$-degree $O(\log\log s(f))$, extending the previously accessible range $d=O(1)$. The proof combines field lifting, multiplicative derivatives, a multiplicity-based generic-rank comparison, a Schur-complement vanishing argument, and dimension-preserving descent to $F_2$ to lower the degree on every coset of a common subspace.