Let T be a bounded linear operator acting on a complex Banach space X and (λ n ) n∈N a sequence of complex numbers. Our main result is that if |λ n |/|λ n+1 | → 1 and the sequence (λ n T n ) n∈N is frequently universal then T is topologically multiply recurrent. To achieve such a result one has to carefully apply Szemerédi's theorem in arithmetic progressions. We show that the previous assumption on the sequence (λ n ) n∈N is optimal among sequences such that |λ n |/|λ n+1 | converges in [0, ∞]. In the case of bilateral weighted shifts and adjoints of multiplication operators we provide characterizations of topological multiple recurrence in terms of the weight sequence and the symbol of the multiplication operator respectively.

Abstract. We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions. The most studied notion in linear dynamics is that of hypercyclicity: a bounded linear operator T acting on a separable Banach space is hypercyclic if there exists a vector whose orbit under T is dense in the space. On the other hand, a very central notion in topological dynamics is that of recurrence. This notion goes back to Poincaré and Birkhoff and it refers to the existence of points in the space for which parts of their orbits under a continuous map "return" to themselves. The purpose of this note is the study of the notion of recurrence, together with its variations, in the context of linear dynamics. Some examples and characterizations of recurrence for special In an effort to characterize recurrent linear operators one many times falls back to the notion of hypercyclicity. This is for example the case when we study the recurrence properties of backwards shifts, say on ℓ 2 (Z). The reason behind is that, according to a result of Seceleanu, [51], the orbits of these operators satisfy a zero-one law: if the orbit of a weighted backward shift contains a non-zero limit point then the corresponding shift is actually hypercyclic. Thus a weighted backward shift on ℓ 2 (Z) is recurrent if and only if it is hypercyclic. The same equivalence is true, albeit for different reasons, for the adjoint of a multiplication operators on the Hardy space H 2 (D). These connections to hypercyclicity, already observed in [17], come up naturally and thus motivate a further search on whether the properties of recurrent operators resemble the properties of hypercyclic ones, in general. It turns out that, indeed, there are many structural similarities between the set of hypercyclic vectors and the set of recurrent vectors in the sense that they exhibit the same invariances. Furthermore, the spectral properties of hypercyclic and recurrent operators are somewhat similar, although this vague statement should be interpreted with some care. However, these similarities cannot be pushed too much as there are obviously many classes of operators which are recurrent without being hypercyclic. One can find such examples among composition operators on the Hardy space H 2 (D). However, the primordial example is given just by considering unimodular multiples of the identity operator. A more general class for which one needs to address the recurrence properties independently of hypercyclicity is that of unitary operators on Hilbert spaces.The discussion above hopefully justifies why we will shortly recall a full set of definitions rel...

Abstract. Let A N be an N-point set in the unit square and consider the Discrepancy function, and |[ 0, x)| denotes the Lebesgue measure of the rectangle. We give various refinements of a well-known result of (Schmidt, 1972) The case of α = ∞ is the Theorem of Schmidt. This estimate is sharp. For the digit-scrambled van der Corput sequence, we havewhenever N = 2 n for some positive integer n. This estimate depends upon variants of the Chang-Wilson-Wolff inequality (Chang et al., 1985). We also provide similar estimates for the BMO norm of D N .

Let B be a homothecy invariant collection of convex sets in R n . Given a measure µ, the associated weighted geometric maximal operator M B,µ is defined by

Abstract. Composition of Web Services (WSs) is anticipated as the future standard way to dynamically build distributed applications, and hence their verification and validation is attracting great attention. The standardization of BPEL as a composition language and of WSDL as a WS interface definition language has led researchers to investigate verification and validation techniques mainly focusing on the sequence of events in the composition, while minor attention has been paid to the validation of the data flow exchange. In this chapter we study the potential of using data flow modelling for testing composite WSs. After an exhaustive exploration of the issues on testing based on data-related models, we schematically settle future research issues on the perspectives opened by data flow-based validation and present examples for some of them, illustrated on the case study of a composite WS that we have developed, the Virtual Scientific Book Store.