Christoph Aistleitner

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is called the star-discrepancy of (z1, . . . , zN ). Here and in the sequel λ denotes the sdimensional Lebesgue measure. The Koksma-Hlawka inequality states that the difference between the integral of a function f over the s-dimensional unit cube and the arithmetic mean of the function values f(z1), . . . , f(zN ) is bounded by the product of the total(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)
By a classical result of Weyl (1916), for any increasing sequence (nk) of positive integers, (nkx) 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 nk = k (Khinchin (1924)) and for lacunary (nk) (Philipp (1975)). In this paper we extend Philipp’s result to(More)
By a profound result of Heinrich, Novak, Wasilkowski, and Woźniakowski the inverse of the stardiscrepancy n∗(s, ε) satisfies the upper bound n∗(s, ε) ≤ cabssε. This is equivalent to the fact that for any N and s there exists a set of N points in [0, 1] whose star-discrepancy is bounded by cabss 1/2N−1/2. The proof is based on the observation that a random(More)
We establish a connection between the L norm of sums of dilated functions whose jth Fourier coefficients are O(j−α) for some α ∈ (1/2, 1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L and for the almost everywhere convergence of(More)
Let f(x) be a 1-periodic function of bounded variation having mean zero, and let (nk)k≥1 be an increasing sequence of positive integers. Then a result of Baker implies the upper bound ∣∣∣∑Nk=1 f(nkx)∣∣∣ = O (√N(logN)3/2+ε) for almost all x ∈ (0, 1) in the sense of the Lebesgue measure. We show that the asymptotic order of ∣∣∣∑Nk=1 f(nkx)∣∣∣ is closely(More)
A classical result of Philipp (1975) states that for any sequence (nk)k≥1 of integers satisfying the Hadamard gap condition nk+1/nk ≥ q > 1 (k = 1, 2, . . .), the discrepancy DN of the sequence (nkx)k≥1 mod 1 satisfies the law of the iterated logarithm (LIL), i.e. 1/4 ≤ lim supN→∞NDN (nkx)(N log logN)−1/2 ≤ Cq a.e. The value of the limsup is a long standing(More)
In 1975 Philipp showed that for any increasing sequence (nk) of positive integers satisfying the Hadamard gap condition nk+1/nk > q > 1, k ≥ 1, the discrepancy DN of (nkx) mod 1 satisfies the law of the iterated logarithm 1/4 ≤ lim sup N→∞ NDN(nkx)(N log logN) −1/2 ≤ Cq a.e. Recently, Fukuyama computed the value of the lim sup for sequences of the form nk =(More)