Sub-linear Upper Bounds on Fourier dimension of Boolean Functions in terms of Fourier sparsity

TL;DR

This paper establishes sub-linear upper bounds on the Fourier dimension of Boolean functions in terms of their sparsity, improving understanding of their spectral complexity and decision tree complexity.

Contribution

The paper proves new upper bounds on Fourier dimension based on sparsity, and connects Fourier dimension to non-adaptive parity decision tree complexity.

Findings

Fourier dimension is at most O(s^{2/3}) for Boolean functions with sparsity s.

Assuming a conjecture, the bound improves to O(\u221a{s}) with polylogarithmic factors.

Fourier dimension and sparsity are quadratically separated in the address function.

Abstract

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of Tsang~\etal~\cite{tsang}, that for every Boolean function of sparsity $s$ there is an affine subspace of $\mathbb{F}_2^n$ of co-dimension $O(\poly\log s)$ restricted to which the function is constant. This conjectured bound is tight upto poly-logarithmic factors as the Fourier dimension and sparsity of the address function are quadratically separated. We obtain these bounds by observing that the Fourier dimension of a Boolean function is equivalent to its non-adaptive parity decision tree complexity, and then bounding the latter.