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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.

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

[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).

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

Strauch & Tóth [1] 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 $0\le\underline d\le\underline\delta\le\overline\delta\le\overline d\le 1.$

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

An ideal result would combine all three density types with the inequality $$ 0\le\underline{u}\le\underline{d}\le\underline{\delta}\le\overline{\delta}\le\overline{d}\le\overline{u}\le1 $$ 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.

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

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.

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

[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).

Strauch & Tóth [1] 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 $0\le\underline d\le\underline\delta\le\overline d\le\overline\delta\le1.$

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

An ideal result would combine all three density types with the inequality $$ 0\le\underline{u}\le\underline{d}\le\underline{\delta}\le\overline{\delta}\le\overline{d}\le\overline{u}\le1 $$ 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.

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

Strauch & Tóth [1] 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 $0\le\underline d\le\underline\delta\le\overline\delta\le\overline d\le 1.$

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

An ideal result would combine all three density types with the inequality $$ 0\le\underline{u}\le\underline{d}\le\underline{\delta}\le\overline{\delta}\le\overline{d}\le\overline{u}\le1 $$ 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.

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