Given $\mathscr{ B}\subseteq \mathbb{ N}$, let $\mathcal{ M}_\mathscr{ B}=\bigcup_{ b\in\mathscr{ B}}b\mathbb{ Z}$ be the correspoding set of multiples. We say that $\mathscr{ B}$ is taut if the logarithmic density of $\mathcal{ M}_\mathscr{ B}$ decreases after removing any element from $\mathscr{ B}$. We say that $\mathscr{ B}$ is minimal if it is primitive (i.e.\ $b| b'$ for $b,b'\in\mathscr{ B}$ implies $b=b'$) and the characteristic function $\eta$ of $\mathcal{ M}_\mathscr{ B}$ is a Toeplitz sequence (i.e.\ for every $n\in \mathbb{ N}$ there exists $s_n$ such that $\eta$ is constant along $n+s_n\mathbb{ Z}$). With every $\mathscr{ B}$ one associates the corresponding taut set $\mathscr{ B}'$ (determined uniquely among all taut sets by the condition that the associated Mirsky measures agree) and the minimal set $\mathscr{ B}^*$ (determined uniquely among all minimal sets by the condition that every configuration appearing on $\mathcal{ M}_{ \mathscr{ B}^*}$ appears on $\mathcal{ M}_\mathscr{ B}$: for every $n\in \mathbb{ N}$, there exists $k\in \mathbb{ Z}$ such that $\mathcal{ M}_{ \mathscr{ B}^*}\cap [0,n]=\mathcal{ M}_\mathscr{ B} \cap[k,k+n]-k$). Besicovitch [2] gave an example of $\mathscr{ B}$ whose set of multiples does not have the natural density. It was proved in [7, Lemma 4.18] that if $\mathcal{ M}_{ \mathscr{ B}'}$ posses the natural density then so does $\mathcal{ M}_\mathscr{ B}$. In this paper we show that this is the only obstruction: every configuration $ijk\in \{ 0,1\}^3$ (with $ij\neq 01$), encoding the information on the existence of the natural density for the triple $\mathcal{ M}_\mathscr{ B},\mathcal{ M}_{ \mathscr{ B'}},\mathcal{ M}_{ \mathscr{ B}^*}$, can occur. Furthermore, we show that $\mathcal{ M}_\mathscr{ B}$ and $\mathcal{ M}_{ \mathscr{ B}'}$ can differ along a set of positive upper density.
Categories: math.DS math.NT