~~Strauch & Tóth [1]~~ Georges Grekos [3][4] showed that for any choice of upper and lower density, there is some subset of $\mathbb{N}$ with the chosen densities, provided the lower is no more than the upper and both are in [0, 1].

Mišík [2] extended this to show that for any choice of upper density, lower density, upper logarithmic density, and lower logarithmic density there is a subset of $\mathbb{N}$ with the chosen densities, provided they follow the necessary inequalities $0\leqslant\underline d\leqslant\underline\delta\leqslant\overline\delta\leqslant\overline d\leqslant 1.$

Is there a similar result with the uniform densities? $$ \underline{u}(A)=\lim_s\frac1s\liminf_n\sum_\stackrel{a\in A}{\ n < a\leqslant n+s}1 $$ $$ \overline{u}(A)=\lim_s\frac1s\limsup_n\sum_\stackrel{a\in A}{\ n < a\leqslant n+s}1 $$

An ideal result would combine all three density types with the inequality $$ 0\leqslant\underline{u}\leqslant\underline{d}\leqslant\underline{\delta}\leqslant\overline{\delta}\leqslant\overline{d}\leqslant\overline{u}\leqslant1 $$ but I'm looking for any published results on the topic.

[1] Strauch and Tóth, Asymptotic density of $A\subset N$ and density of the ratio set $R(A)$. *Acta Arith.* **87** (1998), pp. 67-78. eudml

[2] Ladislav Mišík, Sets of positive integers with prescribed values of densities, *Mathematica Slovaca* **52**:3 (2002), pp. 289-296.dml.cz

[3] Georges Grekos, Sur la répartition des densités des sous-suites d'une suite donnée, PhD thesis, Université Pierre et Marie Curie (1976).

[4] Georges Grekos, Répartition des densites des sous-suites d'une suite d'entiers, *J. Number Theory* **10**:2 (1978), pp. 177-191 (in French). doi:10.1016/0022-314X(78)90034-3

veryoccasionally (e.g. introduce a segment of length $n$ at location $e^{e^n}$) introduce lengthening segments with very uniform density $\bar u$ and $\underline u$ (e.g. obtained from an irrational rotation). This would mean you're done provided you start off with a sequence where $\underline u=\underline d$ and $\bar d=\bar u$. Thus provided Mi\v sik satisfies this, there's nothing more to check. $\endgroup$Répartition Des Densites Des Sous-Suites D'Une Suite D'Entiers, J. Number Theory10(1978), No. 2, 177-191 (in French). Strauch & Tóth's paper is definitely focused on a different problem (what's the minimal lower asymptotic density of a set $X\subseteq\bf N^+$ for which the ratio set of $X$, viz. $\{x/y: x,y\in X\}$, is dense in $[0,\infty[$?), and recovers Grekos' result as a corollary. $\endgroup$Sets with prescribed arithmetic densities, Unif. Distr. Theory3(2008), No. 2, 67-80, where no reference is made to any gap or other issue in Mišík's work). $\endgroup$5more comments