Local one-side porosity and pretangent spaces

TL;DR

This paper explores the relationship between local porosity measures of subsets of positive real numbers and their associated pretangent spaces, extending the concept using a scaling function for subsets of natural numbers.

Contribution

It introduces a novel approach linking porosity numbers with the geometry of pretangent spaces through a scaling function, expanding the understanding of local porosity.

Findings

Porosity numbers are connected to the geometry of pretangent spaces.

Scaling functions extend porosity concepts to subsets of natural numbers.

Main results describe interconnections between porosity, scaling functions, and pretangent space features.

Abstract

For subsets of $\mathbb{R}^+$ we consider the local right upper porosity and the local right lower porosity as elements of a cluster set of all porosity numbers. The use of a scaling function $\mu:\mathbb{N} \to \mathbb{R}^+$ provides an extension of the concept of porosity numbers on subsets of $\mathbb{N}$. The main results describe interconnections between porosity numbers of a set, features of the scaling functions and the geometry of so-called pretangent spaces to this set.