On a set of asymptotic densities

Jahoda, Pavel, and Jahodová, Monika. "On a set of asymptotic densities." Acta Mathematica Universitatis Ostraviensis 16.1 (2008): 21-30. <http://eudml.org/doc/35173>.

@article{Jahoda2008,
abstract = {Let $\mathbb \{P\} = \lbrace p_1, p_2, \dots , p_i, \dots \rbrace $ be the set of prime numbers (or more generally a set of pairwise co-prime elements). Let us denote $A_p^\{a,b\} = \lbrace p^\{an+b\}m \mid n \in \mathbb \{N\} \cup \lbrace 0\rbrace ;m \in \mathbb \{N\}, p \mathrm \{\, does \, not \, divide \,\} m \rbrace $, where $a \in \mathbb \{N\}, b \in \mathbb \{N\} \cup \lbrace 0\rbrace $. Then for arbitrary finite set $B$, $B \subset \mathbb \{P\}$ holds \[d\left(\bigcap \_\{p\_i \in B\} A\_\{p\_i\}^\{a\_i,b\_i\} \right) = \prod \_\{p\_i \in B\} d \left(A\_\{p\_i\}^\{a\_i,b\_i\}\right),\] and \[d \left(A\_\{p\_i\}^\{a\_i,b\_i\}\right) = \frac\{\frac\{1\}\{p\_\{i\}^\{b\_i\}\}\left(1 - \frac\{1\}\{p\_i\}\right)\}\{1 - \frac\{1\}\{p\_\{i\}^\{a\_i\}\}\}.\] If we denote \[A = \left\lbrace \frac\{\frac\{1\}\{p^b\}\left(1 - \frac\{1\}\{p\}\right)\}\{1 - \frac\{1\}\{p^a\}\} \mid p \in \mathbb \{P\}, a \in \mathbb \{N\}, b \in \mathbb \{N\} \cup \lbrace 0\rbrace \right\rbrace ,\] where $\mathbb \{P\}$ is the set of all prime numbers, then for closure of set $A$ holds \[\mathop \{\rm cl\}A = A \cup B \cup \lbrace 0, 1\rbrace ,\] where $B = \left\lbrace \frac\{1\}\{p^b\}\left(1 - \frac\{1\}\{p\}\right) \mid p \in \mathbb \{P\}, b \in \mathbb \{N\} \cup \lbrace 0\rbrace \right\rbrace $.},
author = {Jahoda, Pavel, Jahodová, Monika},
journal = {Acta Mathematica Universitatis Ostraviensis},
keywords = {asymptotic density; density; asymptotic density; density},
language = {eng},
number = {1},
pages = {21-30},
publisher = {University of Ostrava},
title = {On a set of asymptotic densities},
url = {http://eudml.org/doc/35173},
volume = {16},
year = {2008},
}

TY - JOUR
AU - Jahoda, Pavel
AU - Jahodová, Monika
TI - On a set of asymptotic densities
JO - Acta Mathematica Universitatis Ostraviensis
PY - 2008
PB - University of Ostrava
VL - 16
IS - 1
SP - 21
EP - 30
AB - Let $\mathbb {P} = \lbrace p_1, p_2, \dots , p_i, \dots \rbrace $ be the set of prime numbers (or more generally a set of pairwise co-prime elements). Let us denote $A_p^{a,b} = \lbrace p^{an+b}m \mid n \in \mathbb {N} \cup \lbrace 0\rbrace ;m \in \mathbb {N}, p \mathrm {\, does \, not \, divide \,} m \rbrace $, where $a \in \mathbb {N}, b \in \mathbb {N} \cup \lbrace 0\rbrace $. Then for arbitrary finite set $B$, $B \subset \mathbb {P}$ holds \[d\left(\bigcap _{p_i \in B} A_{p_i}^{a_i,b_i} \right) = \prod _{p_i \in B} d \left(A_{p_i}^{a_i,b_i}\right),\] and \[d \left(A_{p_i}^{a_i,b_i}\right) = \frac{\frac{1}{p_{i}^{b_i}}\left(1 - \frac{1}{p_i}\right)}{1 - \frac{1}{p_{i}^{a_i}}}.\] If we denote \[A = \left\lbrace \frac{\frac{1}{p^b}\left(1 - \frac{1}{p}\right)}{1 - \frac{1}{p^a}} \mid p \in \mathbb {P}, a \in \mathbb {N}, b \in \mathbb {N} \cup \lbrace 0\rbrace \right\rbrace ,\] where $\mathbb {P}$ is the set of all prime numbers, then for closure of set $A$ holds \[\mathop {\rm cl}A = A \cup B \cup \lbrace 0, 1\rbrace ,\] where $B = \left\lbrace \frac{1}{p^b}\left(1 - \frac{1}{p}\right) \mid p \in \mathbb {P}, b \in \mathbb {N} \cup \lbrace 0\rbrace \right\rbrace $.
LA - eng
KW - asymptotic density; density; asymptotic density; density
UR - http://eudml.org/doc/35173
ER -