Realisability of simultaneous density constraints for sets of integers

TL;DR

This paper characterizes the possible density quadruplets for subsets of natural numbers, showing they form a set with non-empty interior and positive measure, using probabilistic and diophantine methods.

Contribution

It completes the description of the set of density quadruplets and proves it has positive measure, extending previous results with new methods.

Findings

The set of all possible quadruplets has non-empty interior.

The set of quadruplets has positive measure.

Explicit descriptions of the projections on coordinate planes.

Abstract

In this note, we study the set $\mathcal{D}$ of values of the quadruplet $(\underline{\mathrm{d}}(A),\overline{\mathrm{d}}(A),\underline{\mathrm{d}}(2A),\overline{\mathrm{d}}(2A))$ where $A\subset\mathbb{N}$ and $\underline{\mathrm{d}},\overline{\mathrm{d}}$ 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.