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

Abstract

Abstract. For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of { αa(x) } x≥1 is asymptotically Poissonian for almost all α 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 α such that { αx d } fails to have Poissonian pair correlation is at most 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 2 d+1 .

Pair correlations and equidistribution

“…In our setting, when (x n ) n≥1 are real numbers in the unit interval, we define a function F N (s) as (1) F N (s) = 1 N 1 ≤ m, n ≤ N, m = n : x m − x n ≤ s N , and F (s) = lim N →∞ F N (s), provided that such a limit exists; here s ≥ 0 is a real number, and · denotes the distance to the nearest integer. The function F N (s) counts the number of pairs (x m , x n ), 1 ≤ m, n ≤ N, m = n, of points which are within distance at most s/N of each other (in the sense of the distance on the torus).…”

Equidistribution results for sequences of polynomials

“…Much of the recent interest surrounding whether a sequence (a n α) ∞ n=1 has Poissonian pair correlations comes from a connection with additive combinatorics, and more specifically with the so called additive energy of a sequence (a n ) ∞ n=1 . This connection was initially observed by Aistleitner et al in [4] and subsequently pursued by several authors. For more on this connection we refer the reader to the survey of Larcher and Stockinger [17] and the references therein.…”

7. On pair correlation of sequences

“…, a N ) ≤ N 3 always holds. In [6] the following was shown: Theorem 6. Let (a n ) n≥1 be a strictly increasing sequence of integers such that there exists ε > 0 with Example 3.…”

“…Recently, a much more general metric result on PPC of sequences of the form ({a n α}) n≥1 was given in [6] which shows that there is an intimate connection between the concept of PPC of sequences ({a n α}) n≥1 and the notion of additive energy of the sequence (a n ) n≥1 . The concept of additive energy plays a central role in additive combinatorics and also appears in the study of the metrical discrepancy theory of sequences ({a n α}) n≥1 (see [2,5]).…”

“…, a N ) = Ω(N 3 ) that the sequence ({a n α}) n≥1 has PPC for almost all α? Both questions were answered by J. Bourgain in an Appendix to [6].…”