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 $wt(f)$.
翻译:Chang's lemmma (Duke Mathematical Journal, 2002) 是数学和计算机科学中若干领域应用的经典参数。 对于布尔函数, 其值在 {-1, 1} 让美元(f) 表示它的 Fleier 排名。 对于每个正下限 $t 美元, 张的lemmma 提供较低的约束值在 $wt (f) :\\\ f[x) =-1美元 上方字符的大小。 Freyer 系数至少为 1美元(x) 。 我们的Freyer 系数在数学和计算机科学上方值的大小(S) 更低的参数。 我们的利玛值在 limmma w. t. t. tret.