Realisability of simultaneous density constraints for sets of integers

In this note, we study the set $\mathcal{D}$ of values of the quadruplet $(\underset{‾}{\mathrm{d}}(A),\bar{\mathrm{d}}(A),\underset{‾}{\mathrm{d}}(2A),\bar{\mathrm{d}}(2A))$ where $A\subset \mathbb{N}$ and $\underset{‾}{\mathrm{d}},\bar{\mathrm{d}}$ denote the lower and upper asymptotic density, respectively. Completing existing results on the topic, we determine each of its six projections on coordinate planes, that is, the sets of possible values of the six subpairs of the quadruplet. Further, we show that this set $\mathcal{D}$ has non empty interior, in particular has positive measure. To do so, we use among others probabilistic and diophantine methods. Some auxiliary results pertaining to these methods may be of general interest.