An inequality for the Fourier spectrum of parity decision trees

TL;DR

This paper establishes a new bound on the Fourier spectrum of Boolean functions with respect to parity decision trees, leading to insights into their complexity and specific lower bounds for recursive majority functions.

Contribution

It introduces a novel inequality relating Fourier coefficients to parity decision tree complexity, extending previous work on regular decision trees.

Findings

New bound on Fourier coefficients in terms of parity decision tree complexity

First non-trivial lower bound on parity decision tree complexity of recursive majority

Generalization of O'Donnell and Servedio's inequality to parity decision trees

Abstract

We give a new bound on the sum of the linear Fourier coefficients of a Boolean function in terms of its parity decision tree complexity. This result generalizes an inequality of O'Donnell and Servedio for regular decision trees. We use this bound to obtain the first non-trivial lower bound on the parity decision tree complexity of the recursive majority function.