  The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm
of the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is bounded
by $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot
\|_\infty$ denotes the supremum norm on the polydisc $\D^n$. The main result of
this paper is that this inequality is hypercontractive, i.e., the constant can
be taken to be $C^m$ for some $C>1$. Combining this improved version of the
Bohnenblust--Hille inequality with other results, we obtain the following: The
Bohr radius for the polydisc $\D^n$ behaves asymptotically as $\sqrt{(\log
n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant
for the set of frequencies $\bigl\{\log n: n \text{a positive integer} \le
N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as
$N\to \infty$.
