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.

We study the discrepancy of jittered sampling sets: such a set P ⊂ [0, 1] d is generated for fixed m ∈ N by partitioning [0, 1] d into m d axis aligned cubes of equal measure and placing a random point inside each of the N = m d cubes. We prove that, for N sufficiently large, 1 10where the upper bound with an unspecified constant C d was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N . Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N d d . We also prove a partition principle showing that every partition of [0, 1] d combined with a jittered sampling construction gives rise to a set whose expected squared L 2 −discrepancy is smaller than that of purely random points.

a b s t r a c tMotivated by recent ideas of Harman (Unif. Distrib. Theory, 2010) we develop a new concept of variation of multivariate functions on a compact Hausdorff space with respect to a collection D of subsets. We prove a general version of the Koksma-Hlawka theorem that holds for this notion of variation and discrepancy with respect to D. As special cases, we obtain Koksma-Hlawka inequalities for classical notions, such as extreme or isotropic discrepancy. For extreme discrepancy, our result coincides with the usual Koksma-Hlawka theorem. We show that the space of functions of bounded D-variation contains important discontinuous functions and is closed under natural algebraic operations. Finally, we illustrate the results on concrete integration problems from integral geometry and stereology.

The recently introduced concept of D-variation unifies previous concepts of variation of multivariate functions. In this paper, we give an affirmative answer to the open question from [20] whether every function of bounded Hardy-Krause variation is Borel measurable and has bounded D-variation. Moreover, we show that the space of functions of bounded D-variation can be turned into a commutative Banach algebra.

A permuted van der Corput sequence $S_b^\sigma$ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. $t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$ is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for $t\left({S_p^\sigma } \right)$ for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that $t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$ .