  We establish a generic form of hardness amplification for the approximability
of constant-depth Boolean circuits by polynomials. Specifically, we show that
if a Boolean circuit cannot be pointwise approximated by low-degree polynomials
to within constant error in a certain one-sided sense, then an OR of disjoint
copies of that circuit cannot be pointwise approximated even with very high
error. As our main application, we show that for every sequence of degrees
$d(n)$, there is an explicit depth-three circuit $F: \{-1,1\}^n \to \{-1,1\}$
of polynomial-size such that any degree-$d$ polynomial cannot pointwise
approximate $F$ to error better than
$1-\exp\left(-\tilde{\Omega}(nd^{-3/2})\right)$. As a consequence of our main
result, we obtain an $\exp\left(-\tilde{\Omega}(n^{2/5})\right)$ upper bound on
the the discrepancy of a function in AC$^0$, and an
$\exp\left(\tilde{\Omega}(n^{2/5})\right)$ lower bound on the threshold weight
of AC$^0$, improving over the previous best results of
$\exp\left(-\Omega(n^{1/3})\right)$ and $\exp\left(\Omega(n^{1/3})\right)$
respectively.
  Our techniques also yield a new lower bound of
$\Omega\left(n^{1/2}/\log^{(d-2)/2}(n)\right)$ on the approximate degree of the
AND-OR tree of depth $d$, which is tight up to polylogarithmic factors for any
constant $d$, as well as new bounds for read-once DNF formulas. In turn, these
results imply new lower bounds on the communication and circuit complexity of
these classes, and demonstrate strong limitations on existing PAC learning
algorithms.
