Boolean Functions with Small Approximate Spectral Norm

TL;DR

This paper addresses the structure of Boolean functions on the Boolean cube with small approximate spectral norm by introducing a $k$-affine connectivity framework for a set $A$ and its complement. It proves that if $\|\mathbf{1}_A\|_{A,\epsilon}$ is small and both $A$ and $A^c$ are $k$-affine connected, then $A$ lies in a ring of sets generated by a bounded number of cosets, with a tower-type bound on the coset complexity; if the connectivity fails, the algebra norm is large. The main result extends Cohen's idempotent theorem to approximate norms and provides a new proof for the finite–group case in $\mathbb{F}_2^n$, clarifying the role of approximate norms in controlling additive structure. The findings have implications for learning, pseudorandomness, and complexity theory by linking approximate spectral properties to explicit combinatorial structure of the support.

Abstract

The sum of the absolute values of the Fourier coefficients of a function $f:\mathbb{F}_2^n \to \mathbb{R}$ is called the spectral norm of $f$. Green and Sanders' quantitative version of Cohen's idempotent theorem states that if the spectral norm of $f:\mathbb{F}_2^n \to \{0,1\}$ is at most $M$, then the support of $f$ belongs to the ring of sets generated by at most $\ell(M)$ cosets, where $\ell(M)$ is a constant that only depends on $M$. We prove that the above statement can be generalized to \emph{approximate} spectral norms if and only if the support of $f$ and its complement satisfy a certain arithmetic connectivity condition. In particular, our theorem provides a new proof of the quantitative Cohen's theorem for $\mathbb{F}_2^n$.

Theorems & Definitions (36)

• Proposition 1: Folklore, see Green_Sanders
• Theorem 1: Quantitative Cohen's theorem for $\mathbb{F}_2^n$ MR133397Green_Sanders
• Definition 1: Affine connectivity
• Remark 1
• Theorem 2: Main theorem
• Remark 2
• Remark 3
• Remark 4
• Theorem 3: Cohen's idempotent theorem MR133397
• Lemma 1