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

TL;DR

This paper investigates the conditions under which the fractional parts of sequences exhibit Poissonian pair correlation, linking additive energy estimates to the Hausdorff dimension of exceptional sets, and improves understanding of their size and structure.

Contribution

It establishes a connection between additive energy bounds and Poissonian pair correlation for sequences, providing new estimates for the Hausdorff dimension of exceptional sets.

Findings

Poissonian pair correlation holds for almost all α under certain energy conditions

Hausdorff dimension of the exceptional set is at most (d+2)/(d+3)

Exceptional set has Hausdorff dimension at least 2/(d+1)

Abstract

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.