  For a sequence of integers $\{a(x)\}_{x \geq 1}$ we show that the
distribution of the pair correlations of the fractional parts of $\{ \langle
\alpha a(x) \rangle \}_{x \geq 1}$ is asymptotically Poissonian for almost all
$\alpha$ if the additive energy of truncations of the sequence has a power
savings improvement over the trivial estimate. Furthermore, we give an estimate
for the Hausdorff dimension of the exceptional set as a function of the density
of the sequence and the power savings in the energy estimate. A consequence of
these results is that the Hausdorff dimension of the set of $\alpha$ such that
$\{\langle \alpha x^d \rangle\}$ fails to have Poissonian pair correlation is
at most $\frac{d+2}{d+3} < 1$. This strengthens a result of Rudnick and Sarnak
which states that the exceptional set has zero Lebesgue measure. On the other
hand, classical examples imply that the exceptional set has Hausdorff dimension
at least $\frac{2}{d+1}$.
  An appendix by Jean Bourgain was added after the first version of this paper
was written. In this appendix two problems raised in the paper are solved.
