  In this paper we determine the number and typical structure of sets of
integers with bounded doubling. In particular, improving recent results of
Green and Morris, and of Mazur, we show that the following holds for every
fixed $\lambda > 2$ and every $k \geqslant (\log n)^4$: if $\omega \to \infty$
as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$
with $|A| = k$ and $|A + A| \leqslant \lambda k$ are contained in an arithmetic
progression of length $\lambda k/2 + \omega$.
