  Modifying Besicovitch's construction of a set $\mathcal{B}$ of positive
integers whose set of multiples $\mathcal{M}_{\mathcal{B}}$ has no asymptotic
density, we provide examples of such sets $\mathcal{B}$ for which
$\eta:=1_{\mathbb{Z}\setminus\mathcal{M}_{\mathcal{B}}}\in\{0,1\}^{\mathbb{Z}}$
is a Toeplitz sequence. Moreover our construction produces examples, for which
$\eta$ is not only quasi-generic for the Mirsky measure (which has discrete
dynamical spectrum), but also for some measure of positive entropy. On the
other hand, modifying slightly an example from Kasjan, Keller, and Lema\'nczyk,
we construct a set $\mathcal{B}$ for which $\eta$ is an irregular Toeplitz
sequence but for which the orbit closure of $\eta$ in $\{0,1\}^{\mathbb{Z}}$ is
uniquely ergodic.
