Tight Chang's-lemma-type bounds for Boolean functions

TL;DR

This paper derives new tight bounds on the weight of Boolean functions based on Fourier spectrum measures, improving classical Chang's lemma bounds and demonstrating tightness with specific function constructions.

Contribution

It introduces refined lower bounds on Fourier weight in terms of sparsity, max-support-entropy, and max-rank-entropy, surpassing previous bounds and establishing tightness.

Findings

New bounds improve upon Chang's lemma for various Fourier measures.

Constructed functions demonstrate the tightness of the new bounds.

The bounds asymptotically match the actual Fourier weight for certain functions.

Abstract

Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Boolean function $f$ that takes values in {-1,1} let $r(f)$ denote its Fourier rank. For each positive threshold $t$, Chang's lemma provides a lower bound on $wt(f):=\Pr[f(x)=-1]$ in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least $1/t$. We examine the tightness of Chang's lemma w.r.t. the following three natural settings of the threshold: - the Fourier sparsity of $f$, denoted $k(f)$, - the Fourier max-supp-entropy of $f$, denoted $k'(f)$, defined to be $\max \{1/|\hat{f}(S)| : \hat{f}(S) \neq 0\}$, - the Fourier max-rank-entropy of $f$, denoted $k''(f)$, defined to be the minimum $t$ such that characters whose Fourier coefficients are at least $1/t$ in absolute value span a space of dimension $r(f)$. We prove new lower bounds on $wt(f)$ in terms of these measures. One of our lower bounds subsumes and refines the previously best known upper bound on $r(f)$ in terms of $k(f)$ by Sanyal (ToC, 2019). Another lower bound is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of the absolute values of the level-$1$ Fourier coefficients. We also show that Chang's lemma for the these choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions (which are modifications of the Addressing function) witnessing their tightness. Finally we construct Boolean functions $f$ for which - our lower bounds asymptotically match $wt(f)$, and - for any choice of the threshold $t$, the lower bound obtained from Chang's lemma is asymptotically smaller than $wt(f)$.