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In 2001 Heinrich, Novak, Wasilkowski and Wo´zniakowski proved that for every s ≥ 1 and N ≥ 1 there exists a sequence (z 1 ,. .. , z N) of elements of the s-dimensional unit cube such that the star-discrepancy D * N of this sequence satisfies √ s √ N for some constant c independent of s and N. Their proof uses deep results from probability theory and… (More)

By a well known result of Philipp (1975), the discrepancy D N (ω) of the sequence (n k ω) k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n k+1 /n k ≥ q > 1 (k = 1, 2,. . .). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n k), for subexpenen-tially… (More)

By a profound result of Heinrich, Novak, Wasilkowski, and Wo´zniakowski the inverse of the star-discrepancy n * (s, ε) satisfies the upper bound n * (s, ε) ≤ c abs sε −2. This is equivalent to the fact that for any N and s there exists a set of N points in [0, 1] s whose star-discrepancy is bounded by c abs s 1/2 N −1/2. The proof is based on the… (More)

By a classical result of Weyl (1916), for any increasing sequence (n k) of positive integers , (n k x) is uniformly distributed mod 1 for almost all x. The precise asymptotics of the discrepancy of this sequence is known only in a few cases, e.g. for n k = k (Khinchin (1924)) and for lacunary (n k) (Philipp (1975)). In this paper we extend Philipp's result… (More)

The problem of finding the largest empty axis-parallel box amidst a point configuration is a classical problem in computational complexity theory. It is known that the volume of the largest empty box is of asymptotic order 1/n for n → ∞ and fixed dimension d. However, it is natural to assume that the volume of the largest empty box increases as d gets… (More)

In many applications Monte Carlo (MC) sequences or Quasi-Monte Carlo (QMC) sequences are used for numerical integration. In moderate dimensions the QMC method typically yield better results, but its performance significantly falls off in quality if the dimension increases. One class of randomized QMC sequences, which try to combine the advantages of MC and… (More)

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 N →∞ N D N (n k x)(N log log N) −1/2 ≤ C q a.e. The value of the limsup… (More)

We prove the existence of a limit distribution for the normalized normality measure N (E N)/ √ N (as N → ∞) for random binary sequences E N , by this means confirming a conjecture of Alon, Kohayakawa, Mauduit, Moreira and Rödl. The key point of the proof is to approximate the distribution of the normality measure by the exiting probabilities of a… (More)