Low-Degree Fourier Threshold for Random Boolean Functions

TL;DR

This paper determines the threshold degree for which a random Boolean function is uniquely identified by its low-degree Fourier coefficients, revealing a sharp transition around p/2 with a precise window.

Contribution

It establishes the exact threshold window for low-degree Fourier coefficients to determine a random Boolean function, resolving a question posed by Vershynin.

Findings

Below the threshold, multiple functions share the same low-degree coefficients.

Above the threshold, the function is uniquely determined by its low-degree coefficients.

The threshold is approximately p/2 with an O(√(p log p)) window.

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 \[ d \le \frac{p}{2} - \sqrt{\frac{p}{2}\bigl(\log p + \omega(1)\bigr)}, \] 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.