Abstract. For a sequence of integers {a(x)} x≥1 we show that the distribution of the pair correlations of the fractional parts of { αa(x) } x≥1 is asymptotically Poissonian for almost all α if the additive energy of truncations of the sequence has a power savings improvement over the trivial estimate. Furthermore, we give an estimate for the Hausdorff dimension of the exceptional set as a function of the density of the sequence and the power savings in the energy estimate. A consequence of these results is that the Hausdorff dimension of the set of α such that { αx d } fails to have Poissonian pair correlation is at most d+2 d+3 < 1. This strengthens a result of Rudnick and Sarnak which states that the exceptional set has zero Lebesgue measure. On the other hand, classical examples imply that the exceptional set has Hausdorff dimension at least 2 d+1 .
Abstract. In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are numerical integration rules which achieve an exponential convergence of the worst-case integration error. We also investigate the dependence of the worst-case error on the number of variables s, and show various tractability results under certain conditions on weights of the Korobov space. Tractability means that the dependence on s is never exponential, and sometimes the dependence on s is polynomial or there is no dependence on s at all.
Abstract. We say that a sequence (xn) n≥1 in [0, 1) has Poissonian pair correlations iffor all s > 0. In this note we show that if the convergence in the above expression is-in a certain sense-fast, then this implies a small discrepancy for the sequence (xn) n≥1 . As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in [0, 1).
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