Realisability of simultaneous density constraints for sets of integers
Pierre-Yves Bienvenu
TL;DR
This work advances the realisability problem for simultaneous density constraints by studying the density-profile quadruplets $p_A=(\underline{d}(A),\overline{d}(A),\underline{d}(2A),\overline{d}(2A))$ and the ambient set $\mathcal{D}\subset[0,1]^4$. It completes the six coordinate-plane projections, showing $\mathcal{D}_{2,3}=[0,1]^2$ and $\mathcal{D}_{2,4}=\{\alpha/k: \alpha\in\mathcal{U}, k\in\mathbb{N}\}$ with a precise boundary function $f(\alpha)=\min(\tfrac{3}{2}\alpha,\tfrac{1+\alpha}{2})$, while recapitulating known projections $\mathcal{D}_{1,2}=\mathcal{D}_{3,4}=\{(x,y): y\ge x\}$. It further demonstrates the set $\mathcal{D}$ contains a full-dimensional polyhedron by constructing a four-parameter family $G(\alpha_1,\alpha_2,\beta_1,\beta_2)$ realized via a synthesis of lacunary density results, probabilistic methods, and diophantine tools (Weyl equidistribution, Erdős–Turán bounds). The paper also derives three-dimensional projection consequences and discusses limitations and open directions toward a complete description of $\mathcal{D}$, highlighting the rich interaction between additive combinatorics, probability, and diophantine approximation.
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.