Abstract. Upper bounds for GCD sums of the form (gcd(n k , n ℓ )) 2α (n k n ℓ ) α are established, where (n k ) 1≤k≤N is any sequence of distinct positive integers and 0 < α ≤ 1; the estimate for α = 1/2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 1/2. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson-Hunt-type inequality for systems of dilated functions of bounded variation or belonging to Lip 1/2 , a result that in turn settles two longstanding problems on the a.e. behavior of systems of dilated functions: the a.e. growth of sums of the form N k=1 f (n k x) and the a.e. convergence of ∞ k=1 c k f (n k x) when f is 1-periodic and of bounded variation or in Lip 1/2 .

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. Let α ∈ (1/2, 1) be fixed. We prove thatfor all sufficiently large T , where we can choose c α = 0.18(2α−1) 1−α . The same result has already been obtained by Montgomery, with a smaller value for c α . However, our proof, which uses a modified version of Soundararajan's "resonance method" together with ideas of Hilberdink, is completely different from Montgomery's. This new proof also allows us to obtain lower bounds for the measure of those t ∈ [0, T ] for which |ζ(α + it)| is of the order mentioned above.

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Abstract. In this paper we prove a correspondence principle between multivariate functions of bounded variation in the sense of Hardy and Krause and signed measures of finite total variation, which allows us to obtain a simple proof of a generalized Koksma-Hlawka inequality for non-uniform measures. Applications of this inequality to importance sampling in Quasi-Monte Carlo integration and tractability theory are given. Furthermore, we discuss the problem of transforming a low-discrepancy sequence with respect to the uniform measure into a sequence with low discrepancy with respect to a general measure µ, and show the limitations of a method suggested by Chelson.