The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing 1/N . In the d-dimensional case, of course, the order of the mean spacing is 1/N 1 d , and -in our concept-the distance of sequence elements will be measured by the supremum-norm.Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

We say that a sequence (xn) n∈N in [0, 1) has Poissonian pair correlations iffor every s ≥ 0. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence (xn) n∈N of real numbers in [0, 1) having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and in more general LS-sequences of points and digital (t, 1)-sequences. Additionally, this theorem enables us to derive negative pair correlation properties for sequences of the form ({anα}) n∈N , where (an) n∈N is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of α. Second, in this note we study the pair correlation statistics for sequences of the form, xn = {b n α}, n = 1, 2, 3, . . ., with an integer b ≥ 2, where we choose α as the Stoneham number and as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property. Throughout this article {·} denotes the fractional part of a real number. arXiv:1803.05236v2 [math.NT]

In this paper, we derive fully implementable first-order time-stepping schemes for McKean–Vlasov stochastic differential equations, allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretized interacting particle system associated with the McKean–Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second-order moments. In addition, numerical examples are presented which support our theoretical findings.

In this paper, we introduce a fully implementable, adaptive Euler-Maruyama scheme for McKean SDEs with non-globally Lipschitz continuous drifts. We prove moment stability of the discretised processes and a strong convergence rate of 1/2. We present several numerical examples centred around a mean-field model for FitzHugh-Nagumo neurons, which illustrate that the standard uniform scheme fails and that the adaptive scheme shows in most cases superior performance compared to tamed approximation schemes. In addition, we propose a tamed and an adaptive Milstein scheme for a certain class of McKean SDEs.

We show for sequences (an) n∈N of distinct positive integers with maximal order of additive energy, that the sequence ({anα}) n∈N does not have Poissonian pair correlations for any α. This result essentially sharpens a result obtained by J. Bourgain on this topic.