Boolean functions with small spectral norm, revisited

TL;DR

This paper revisits the structure of integer-valued functions with small spectral norm over finite fields, showing they can be expressed as sums of indicator functions of subspaces with bounded complexity.

Contribution

It refines previous arguments to demonstrate that such functions decompose into a bounded sum of subspace indicators, clarifying the structure in the finite field setting.

Findings

Functions with spectral norm ≤ M decompose into ≤ exp(M^{3+o(1)}) subspace indicator sums.

Provides a clearer argument for the structure of functions with small spectral norm.

Extends previous results to the finite field model for better understanding.

Abstract

We show that if $f$ is an integer-valued function with spectral norm at most $M$ then there are subspaces $V_1,\dots,V_L$ and signs $\sigma_1,\dots,\sigma_L \in \{-1,1\}$ such that $f=\sigma_1 1_{V_1} + \dots + \sigma_L 1_{V_L}$ where $L < \exp(M^{3+o(1)})$. This note extracts out the argument from arXiv:1610.07092 to the model setting of $\mathbb{F}_2^n$. The hope is that it will clarify the arguments of that paper.