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

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