Christoph Aistleitner

ON THE LAW OF THE ITERATED LOGARITHM FOR THE DISCREPANCY OF LACUNARY SEQUENCES

A classical result of Philipp (1975) states that for any sequence (n k ) k ≥1 of integers satisfying the Hadamard gap condition n k+1 /n k ≥ q > 1 (k = 1,2,...), the discrepancy D N of the sequence (n k x)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim sup ND N (n k x)(N log log N) -12 ≤ C q a .e. N→∞ The value of the lim sup is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the lim sup for n k = θ k , θ > 1, not necessarily integer. We extend Fukuyamas result to a large class of integer sequences (n k ) characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the lim sup is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

GCD sums from Poisson integrals and systems of dilated functions

Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}] are proved, where

Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

(n_k)_{1 \leq k \leq N}

On the law of the iterated logarithm for permuted lacunary sequences

is any sequence of distinct positive integers and

On the size of the largest empty box amidst a point set

0<\alpha \le 1

On the asymptotic behavior of weakly lacunary series

; the estimate for

PROBABILITY AND METRIC DISCREPANCY THEORY

\alpha=1/2

Probabilistic discrepancy bound for Monte Carlo point sets

solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for

A Central Limit Theorem For Latin Hypercube Sampling With Dependence And Application To Exotic Basket Option Pricing

\alpha=1/2

On the maximal spectral type of a class of rank one transformations

. 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