  Let T be a bounded linear operator acting on a complex Banach space X and
(\lambda_n) a sequence of complex numbers. Our main result is that if
|\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is
frequently universal then T is topologically multiply recurrent. To achieve
such a result one has to carefully apply Szemer\'edi's theorem in arithmetic
progressions. We show that the previous assumption on the sequence (\lambda_n)
is optimal among sequences such that |\lambda_n|/|\lambda_{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.
