Unilateral weighted shifts on $\ell^2$

Given a unilateral shift $B_w$ (determined by a bounded sequence $w$), a sequence $x \in \ell^2$ is "hypercyclic" for $w$ iff the forward iterates of $x$ under $B_w$ are dense in $\ell^2$. We show that it is possible to make the set of $x \in \ell^2$ which are simultaneously hypercyclic for all $w$ in a fixed $W \subseteq \ell^\infty$ arbitrarily complicated by choosing $W$ appropriately.
Comment: 24 pages