Abstract. A deterministic sequence of real numbers in the unit interval is called equidistributed if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonian if the number of pairs x k , x l ∈ (xn) 1≤n≤N which are within distance s/N of each other is asymptotically ∼ 2sN . A randomly generated sequence has both of these properties, almost surely. There seems to be a vague sense that having Poissonian pair correlations is a "finer" property than being equidistributed. In this note we prove that this really is the case, in a precise mathematical sense: a sequence whose asymptotic distribution of pair correlations is Poissonian must necessarily be equidistributed. Furthermore, for sequences which are not equidistributed we prove that the square-integral of the asymptotic density of the sequence gives a lower bound for the asymptotic distribution of the pair correlations.Let (x n ) n≥1 be a sequence of real numbers. We say that this sequence is equidistributed or uniformly distributed modulo one if asymptotically the relative number of fractional parts of elements of the sequence falling into a certain subinterval is proportional to the length of this subinterval. More precisely, we require thatfor all 0 ≤ a ≤ b ≤ 1, where {·} denotes the fractional part. This notion was introduced in the early twentieth century, and received widespread attention after the publication of Hermann Weyl's seminal paperÜber die Gleichverteilung von Zahlen mod. Eins in 1916 [14]. Among the most prominent results in the field are the facts that the sequences (nα) n≥1 and (n 2 α) n≥1 are equidistributed whenever α ∈ Q, and the fact that for any distinct integers n 1 , n 2 , . . . the sequence (n k α) k≥1 is equidistributed for almost all α. We note that when (X n ) n≥1 is a sequence of independent, identically distributed (i.i.d.) random variables having uniform distribution on [0, 1], then by the Glivenko-Cantelli theorem this sequence is almost surely equidistributed. Consequently, in a very vague sense equidistribution can be seen as an indication of "pseudorandom" behavior of a deterministic sequence. For more information on uniform distribution theory, see the monographs [4,7].The investigation of pair correlations can also be traced back to the beginning of the twentieth century, when such quantities appeared in the context of statistical mechanics. In our setting, when (x n ) n≥1 are real numbers in The first two authors are supported by the Austrian Science Fund (FWF), project Y-901.

The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function ψ : N → R for almost all reals x there are infinitely many coprime solutions (a, n) to the inequality |nx − a| < ψ(n), provided that the series ∞ n=1 ψ(n)ϕ(n)/n is divergent. In the present paper we prove that the conjecture is true under the "extra divergence" assumption that divergence of the series still holds when ψ(n) is replaced by ψ(n)/(log n) ε for some ε > 0. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.MSC classification: 11J83, 11K55, 11K60.

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.

Let (an) n be a strictly increasing sequence of positive integers, denote by AN = {an : n ≤ N } its truncations, and let α ∈ [0, 1]. We prove that if the additive energy E (AN ) of AN is in Ω N 3 , then the sequence ( αan ) n of fractional parts of αan does not have Poissonian pair correlations (PPC) for almost every α in the sense of Lebesgue measure. Conversely, it is known that E (AN ) = O N 3−ε , for some fixed ε > 0, implies that ( αan ) n has PPC for almost every α. This note makes a contribution to investigating the energy threshold for E (AN ) to imply this metric distribution property. We establish, in particular, that there exist sequences (an) n withsuch that the set of α for which (αan) n does not have PPC is of full Lebesgue measure. Moreover, we show that for any fixed ε > 0 there are sequences (an) n with E (AN ) = Θ N 3 log(N)(log log N) 1+ε satisfying that the set of α for which the sequence αan n does not have PPC is of full Hausdorff dimension. AddressesThomas Lachmann,

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