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We study the supremum of some random Dirichlet polynomials $D_N\left(t\right)={\sum }_{n=2}^N\epsilon ₙdₙn^{-\sigma -it}$, where (εₙ) is a sequence of independent Rademacher random variables, the weights (dₙ) are multiplicative and 0 ≤ σ < 1/2. Particular attention is given to the polynomials ${\sum }_{n{\in }_{\tau }}\epsilon ₙn^{-\sigma -it}$, ${}_{\tau }=2\le n\le N:P⁺\left(n\right)\le p_{\tau }$, P⁺(n) being the largest prime divisor of n. We obtain sharp upper and lower bounds for the supremum expectation that extend the optimal estimate of Halász-Queffélec, $su{p}_{t\in ℝ}\left|{\sum }_{n=2}^N\epsilon ₙn^{-\sigma -it}\right|\approx \left(N^{1-\sigma }\right)/\left(logN\right)$. The proofs are entirely based on methods of stochastic processes, in particular the metric...

We investigate compactness properties of weighted summation operators $V_{\alpha ,\sigma }$ as mappings from ℓ₁(T) into ${\ell }_q\left(T\right)$ for some q ∈ (1,∞). Those operators are defined by $\left(V_{\alpha ,\sigma }x\right)\left(t\right):=\alpha \left(t\right){\sum }_{s⪰t}\sigma \left(s\right)x\left(s\right)$, t ∈ T, where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for $eₙ\left(V_{\alpha ,\sigma }\right)$, the (dyadic) entropy numbers of $V_{\alpha ,\sigma }$. The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights with α(t)σ(t)...

The aim of this paper is to provide upper bounds for the entropy numbers of summation operators on trees in a critical case. In a recent paper [Studia Math. 202 (2011)] we elaborated a framework of weighted summation operators on general trees where we related the entropy of the operator to those of the underlying tree equipped with an appropriate metric. However, the results were left incomplete in a critical case of the entropy behavior, because this case requires much more involved techniques....