On the complexity of upper frequently hypercyclic vectors

Given a continuous linear operator $T:X\to X$, where $X$ is a topological vector space, let $\mathrm{U}\mathrm{F}\mathrm{H}\mathrm{C}(T)$ be the set of upper frequently hypercyclic vectors, that is, the set of vectors $x\in X$ such that $\{n\in \omega :T^{n}x\in U\}$ has positive upper asymptotic density for all nonempty open sets $U\subseteq X$. It is known that $\mathrm{U}\mathrm{F}\mathrm{H}\mathrm{C}(T)$ is a $G_{\delta \sigma \delta }$-set which is either empty or contains a dense $G_{\delta }$-set. Using a purely topological proof, we improve it by showing that $\mathrm{U}\mathrm{F}\mathrm{H}\mathrm{C}(T)$ is always a $G_{\delta \sigma }$-set. Bonilla and Grosse-Erdmann asked in [Rev. Mat. Complut. $\mathbf{\text{31}}$ (2018), 673–711] whether $\mathrm{U}\mathrm{F}\mathrm{H}\mathrm{C}(T)$ is always a $G_{\delta }$-set. We answer such question in the negative, by showing that there exists a continuous linear operator $T$ for which $\mathrm{U}\mathrm{F}\mathrm{H}\mathrm{C}(T)$ is not a $F_{\sigma \delta }$-set (hence not $G_{\delta }$). In addition, we study the [non-]equivalence between (the ideal versions of) upper frequently hypercyclicity in the product topology and upper frequently hypercyclicity in the norm topology.