  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 $wt(f)$.
