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Szemeredi's theorem, frequent hypercyclicity and multiple recurrence

Let T be a bounded linear operator acting on a complex Banach space X and (λ_n) a sequence of complex numbers. Our main result is that if |λ_n|/|λ_{n+1}| \to 1 and the sequence (λ_n T^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) is optimal among sequences such that |λ_n|/|λ_{n+1}| converges in [0,+\infty]. 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.