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For a sequence of integers $\{a(x)\}_{x \geq 1}$ we show that thedistribution 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 powersavings improvement over the trivial estimate. Furthermore, we give an estimatefor the Hausdorff dimension of the exceptional set as a function of the densityof the sequence and the power savings in the energy estimate. A consequence ofthese results is that the Hausdorff dimension of the set of $\alpha$ such that$\{\langle \alpha x^d \rangle\}$ fails to have Poissonian pair correlation isat most $\frac{d+2}{d+3} < 1$. This strengthens a result of Rudnick and Sarnakwhich states that the exceptional set has zero Lebesgue measure. On the otherhand, classical examples imply that the exceptional set has Hausdorff dimensionat least $\frac{2}{d+1}$.  An appendix by Jean Bourgain was added after the first version of this paperwas written. In this appendix two problems raised in the paper are solved.

Additive Energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems