Low-Degree Fourier Threshold for Random Boolean Functions

Abstract: We study whether a uniformly random Boolean function $f : {-1,1}^p \to {-1,1}$ is determined by its Walsh--Fourier coefficients of degree at most $d$. We show that the threshold lies at $p/2$ up to an $O(\sqrt{p \log p})$ window: if [ d \le \frac{p}{2} - \sqrt{\frac{p}{2}\bigl(\log p + ω(1)\bigr)}, ] then with probability $1-o(1)$ there exists another Boolean function $g \ne f$ with the same degree-$\le d$ coefficients. Conversely, for every fixed $η\in (0,1)$, if [ d \ge \frac{p}{2} + \sqrt{\frac{p}{2}\log\frac{6p}{η^2}}, ] then with probability at least $1-2^{-p}$, the function $f$ is uniquely determined by its degree-$\le d$ coefficients, even among all bounded functions $g : {-1,1}^p \to [-1,1]$. This resolves a question of Vershynin.