Low-Degree Fourier Threshold for Random Boolean Functions

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 $$d \le \frac{p}{2} - \sqrt{\frac{p}{2}\bigl(\log p + \omega(1)\bigr)},$$ 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 $\eta \in (0,1)$, if $$d \ge \frac{p}{2} + \sqrt{\frac{p}{2}\log\frac{6p}{\eta^2}},$$ 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.