A Metric Discrepancy Result With Given Speed

It is known that the discrepancy DN{ kx} of the sequence { kx} satisfies NDN{ kx} = O((log N) (log log N) 1 + ε) a.e. for all ε> 0 , but not for ε= 0. For nk= θk, θ> 1 we have NDN{ nkx} ≦ (Σ θ+ ε) (2 Nlog log N) 1 / 2 a.e. for some 0 0 , but not for ε 0 , there exists a sequence { nk} of positive integers such that NDN{ nkx} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums. © 2016, Akadémiai Kiadó, Budapest, Hungary

GCD sums from Poisson integrals and systems of dilated functions

Upper bounds for GCD sums of the form (Formula Presented) are established, where (nk)1≤k≤N is any sequence of distinct positive integers and 0 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 (Formula Presented)=1 f(nkx) and the a.e. convergence of (Formula Presented)=1 ckf(nkx) when f is 1-periodic and of bounded variation or in Lip1/2. © European Mathematical Society 2015

Strong approximation of lacunary series with random gaps

We investigate the asymptotic behavior of sums (Formula presented.), where f is a mean zero, smooth periodic function on (Formula presented.) and (Formula presented.) is a random sequence such that the gaps (Formula presented.) are i.i.d. Our result shows that, in contrast to the classical Salem–Zygmund theory, the almost sure behavior of lacunary series with random gaps can be described very precisely without any assumption on the size of the gaps. © 2017 Springer-Verlag Wie

Point sets on the sphere S2\mathbb{S}^2 with small spherical cap discrepancy

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order $N^{-1/2}$. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given

Upper and lower class separating sequences for Brownian motion with random argument

info:eu-repo/semantics/publishe

Maximizing Sudler products via Ostrowski expansions and cotangent sums

There is an extensive literature on the asymptotic order of Sudler’s trigonometric product (Formula presented) for fixed or for “typical” values of α. We establish a structural result which for a given α characterizes those N for which PN (α) attains particularly large values. This characterization relies on the coefficients of N in its Ostrowski expansion with respect to α, and allows us to obtain very precise estimates for max (Formula presented) and for (Formula presented) in terms of M, for any c > 0. Furthermore, our arguments give a natural explanation of the fact that the value of the hyperbolic volume of the complement of the figure-eight knot appears generically in results on the asymptotic order of the Sudler product and of the Kashaev invariant.</p