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

Extreme Values of the Riemann Zeta Function on the 1-Line

We prove that there are arbitrarily large values of t such that {equation presented}. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function

On the number of gaps and Poissonian pair correlations

A sequence (xn) on the unit interval is said to have Poissonian pair correlation if #{1≤i≠j≤N:‖xi−xj‖≤s/N}=2sN(1+o(1)) for all reals s&gt;0, as N→∞. It is known that, if (xn) has Poissonian pair correlations, then the number g(n) of different gap lengths between neighboring elements of {x1,…,xn} cannot be bounded along any index subsequence (nt). First, we improve this by showing that, if (xn) has Poissonian pair correlations, then the maximum among the multiplicities of the neighboring gap lengths of {x1,…,xn} is o(n), as n→∞. Furthermore, we show that for every function f:N+→N+ with limn⁡f(n)=∞ there exists a sequence (xn) with Poissonian pair correlations such that g(n)≤f(n) for all sufficiently large n. This answers negatively a question posed by G. Larcher

Circular automata synchronize with high probability

In this paper we prove that a uniformly distributed random circular automaton An of order n synchronizes with high probability (w.h.p.). More precisely, we prove that [Formula presented] The main idea of the proof is to translate the synchronization problem into a problem concerning properties of a random matrix; these properties are then established with high probability by a careful analysis of the stochastic dependence structure among the random entries of the matrix. Additionally, we provide an upper bound for the probability of synchronization of circular automata in terms of chromatic polynomials of circulant graphs

The Duffin-Schaeffer conjecture with extra divergence

The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function ψ:N→R for almost all reals x there are infinitely many coprime solutions (a,n) to the inequality |nx−a|0. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani