Irregular $\mathcal{B}$-free Toeplitz sequences via Besicovitch's construction of sets of multiples without density

Abstract: 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.