There Is No Khintchine Threshold for Metric Pair Correlations

Abstract

We consider sequences of the form (anα) n mod 1, where α ∈ [0, 1] and where (an) n is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all α in the sense of Lebesgue measure, we say that (an)n has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of (an)n. Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterises the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence (an)n having large additive energy which, however, maintains the metric pair correlation property.

Pair correlation and equidistribution on manifolds

“…There has been significant recent progess in characterising the Poisson pair correlation (1.2) for general sequences ξ j = 〈a j α〉, for Lebesgue-almost every α, in terms of the additive energy of the integer coefficients a j ; cf. [2,5] and references therein. Explicit examples for which Poisson pair correlation (1.2) can be established include the fractional part of square-roots, i.e., ξ j = 〈 j 1/2 〉 [6], and directions of points in a shifted Euclidean lattice [7].…”

On a multi-dimensional Poissonian pair correlation concept and uniform distribution

“…The notion Poissonian pair correlation has attracted renewed interest in the last few years, due to its connection to several mathematical fields, such as Diophantine approxmation, additive combinatorics and uniform distribution (see e.g., [1,3,4,9,11,19]). The link between the concept of uniform distribution modulo 1 and Poissonian pair correlation has been studied in the one-dimensional case.…”

GCD sums and sum-product estimates

“…The metric poissonian property is a strong notion of equidistribution for dilates of sequences, motivated by certain concerns in quantum physics (see [14]). We prove the following theorem, continuing the line of work [4,16,7,3] that investigates the relationship between the metric poissonian property and the notion of additive energy. Theorem 6.…”

“…Though in Theorem 6 we have refined the state of knowledge about the relationship between the metric poissonian property and additive energy, it is still unclear what the truth should be. We know from [3] that there is no sharp threshold phenomenon, but we still believe that Theorem 6 is true for any C greater than 1, which, up to (log N) o(1) factors, would be an optimal result.…”

“…We make no attempt to optimise the value of C in Theorem 6. We have previously speculated that any value of C greater than 1 should suffice (Fundamental question, [7]), and this could still be true (it is not precluded by the counterexample constructed in [3]). Unfortunately it does not seem as if the method presented in this paper will be able to prove this result, at least not without substantial modification.…”