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.
Given an irrational number α ∈ (0, 1), the Sudler product is defined by P N (α) = N r=1 2| sin πrα|. Answering a question of Grepstad, Kaltenböck and Neumüller we prove an asymptotic formula for distorted Sudler products when α is the golden ratio ( √ 5 + 1)/2 and establish that in this case lim sup N →∞ P N (α)/N < ∞. We obtain similar results for quadratic irrationals α with continued fraction expansion α = [a, a, a, . . . ] for some integer a ≥ 1, and give a full characterization of the values of a for which lim inf N →∞ P N (α) > 0 and lim sup N →∞ P N (α)/N < ∞ hold, respectively. We establish that there is a (sharp) transition point at a = 6, and resolve as a by-product a problem of the first named author, Larcher, Pillichshammer, Saad Eddin, and Tichy.
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,
A well known result in the theory of uniform distribution modulo one (which goes back to Fejér and Csillag) states that the fractional parts {n α } of the sequence (n α ) n≥1 are uniformly distributed in the unit interval whenever α > 0 is not an integer. For sharpening this knowledge to local statistics, the k-level correlation functions of the sequence ({n α }) n≥1 are of fundamental importance. We prove that for each k ≥ 2, the k-level correlation function R k is Poissonian for almost every α > 4k 2 − 4k − 1.
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