Szemerédi's theorem, frequent hypercyclicity and multiple recurrence (Q2890399)

Let \(T\) be a bounded linear operator acting on a complex Banach space \(X\). The operator \(T\) is called \textit{hypercyclic} (respectively, \textit{frequently hypercyclic}) if there is \(x\in X\) such that the set \(\{T^nx: \;n=0,1,2,\ldots\}\) is dense in \(X\) (respectively, if there is \(x\in X\) such that, for every non-empty open set \(U\) in \(X\), the set \(\{n\in {\mathbb N}: T^nx\in U\}\) has positive lower density). The operator \(T\) is called \textit{recurrent} (respectively, \textit{topologically multiply recurrent}) if, for every non-empty open set \(U\) in \(X\), there is some positive integer \(k\) such that \(U\cap T^{-k}U\not=\emptyset\) (respectively, if for every non-empty open set \(U\) in \(X\) and every positive integer \(m\), there is some positive integer \(k\) such that \(U\cap T^{-k}U\cap \ldots \cap T^{-mk}U\not=\emptyset\)). A sequence \(\{T_n\}_{n\in {\mathbb N}}\) of operators acting on \(X\) is called \textit{frequently universal} if there is \(x\in X\) such that, for every non-empty open set \(U\) in \(X\), the set \(\{n\in {\mathbb N}: T_nx\in U\}\) has positive lower density.NEWLINENEWLINEClearly, every hypercyclic operator is recurrent. But, a hypercyclic operator is not topologically multiply recurrent in general. On the other hand, it is known that an operator \(T\) is hypercyclic whenever the sequence \(\{\frac{1}{n}T^n\}_{n\in {\mathbb N}}\) is frequently universal. Motivated by this, the authors examine in the paper under review when the hypothesis that \(\{\lambda_{n}T^n\}_{n\in {\mathbb N}}\) is frequently universal for some sequence of complex numbers \((\lambda_n)_{n\in {\mathbb N}}\) implies that \(T\) is topologically multiply recurrent. Precisely, they show that, if \(\lim_{n\to\infty}\frac{|\lambda_n|}{|\lambda_{n+\tau}|}=1\) for some positive integer \(\tau\) and the sequence \(\{\lambda_{n}T^n\}_{n\in {\mathbb N}}\) is frequently universal, then \(T\) is topologically multiply recurrent. The authors also show that, for each \(a\in [0,\infty)\setminus\{1\}\) and \(\tau\in {\mathbb N}\), there is a sequence \((\lambda_n)_{n\in {\mathbb N}}\) and an operator \(T\) which is not even recurrent, such that \(\lim_{n\to\infty}\frac{|\lambda_n|}{|\lambda_{n+\tau}|}=a\) and the sequence \(\{\lambda_{n}T^n\}_{n\in {\mathbb N}}\) is frequently universal. The proofs follow by applying carefully Szemerédi's theorem in arithmetic progressions.