On $[\varrho_1,\varrho_2]$-lower superdense sets

Abstract

The main goal of the paper  is to characterize the families
of~$[\varrho_1,\varrho_2]$\discretionary{-}{-}{-}lower superdense subsets of $\mathbb{R}$, which generalize well-known notions
of~density topology (in our paper, denoted by $L^\blacklozenge_1$) and $T^\ast$ topology. Dense sets preserve  density,
i.e.,
if $E\in L^\blacklozenge_1$, then for every  measurable $F\subset \mathbb{R}$,   $E\cap F$  possesses the same lower and upper density as $F$ at every $x\in E\cap F$\!.
On the other hand, elements of~$T^\ast$\,preserve positive lower density, i.e.,
if $E\in T^\ast$\!,\,$F$ is measurable, $x\in E\cap F$ and $\underline{d}\,(F,x)>0$, then $\underline{d}\,(E\cap F,x)>0$.
 Taking arbitrary $0\leq \varrho_1\leq \varrho_2\leq 1$, $\varrho_2-\varrho_1<1$, we can define subsets $E\subset \mathbb{R}$ which preserve $[\varrho_1,\varrho_2]$-lower density,
 i.e.,
 if $F$ is measurable, $x\in E\cap F$ and $\underline{d}\,(F,x)$ is greater or not less than  $\varrho_2$, then $\underline{d}\,(E\cap F,x)$ is greater or not less than $\varrho_1$.
 We can define four types of superdense sets, but three of them are equal.
 Even though the definition and properties of~$[\varrho_1,\varrho_2]$-lower superdensity and $T^\ast$\,topology are similar and all of them consist
 of very big sets, these families  are essentially different.
 In the paper, we focus on~basic properties, characterizations
 of superdense sets and relationships between
 $[\varrho_1,\varrho_2]$-lower superdense sets for different indices $[\varrho_1,\varrho_2]$.
We apply the notion of~$[\varrho_1,\varrho_2]$-lower superdensity
to find adders of $\varrho$-lower continuous functions.