Disjoint Hypercyclicity along filters

We extend a result of B`{e}s, Martin, Peris and Shkarin by stating: $B_w$ is $\mathcal{F}$-weighted backward shift if and only if $(B_w,\dots ,B_w^r)$ is $d$-$\mathcal{F}$, for any $r\in \mathbb{N}$, where $\mathcal{F}$ runs along some filters containing strictly the family of cofinite sets, which are frequently used in Ramsey theory. On the other hand, we point out that this phenomenon does not occur beyond the weighted shift frame by showing a mixing linear operator $T$ on a Hilbert space such that the tuple $(T,T^2)$ is not $d$-syndetic. We also investigate the relationship between reiteratively hypercyclic operators and $d$-$\mathcal{F}$ tuples, for filters $\mathcal{F}$ contained in the family of syndetic sets. Finally, we examine conditions to impose in order to get reiterative hypercyclicity from syndeticity in the weighted shift frame.