Christoph Aistleitner

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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 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)
• Randomness of sequences • Aperiodic and periodic correlation of sequences • Combinatorial aspects of sequences, including difference sets • Sequences with applications in coding theory and cryptography • Sequences over finite fields/rings/function fields • Linear and nonlinear feedback shift register sequences • Sequences for radar, synchronization, 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 (EN )/ √ N (as N → ∞) for random binary sequences EN , 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)