On Fractionally Dense Sets

TL;DR

This paper proves that certain subsets of natural numbers and non-zero ideals in imaginary quadratic fields are densely distributed in positive real and complex numbers respectively, revealing their fractional density properties.

Contribution

It establishes fractional density of specific subsets of natural numbers and ideals in imaginary quadratic fields, extending understanding of their distribution in real and complex spaces.

Findings

Subsets of natural numbers are fractionally dense in positive reals.

Non-zero ideals in imaginary quadratic fields are fractionally dense in complex numbers.

The results connect algebraic structures with topological density properties.

Abstract

In this article, we prove some subsets of the set of natural numbers $\mathbb{N}$ and any non-zero ideals of an order of imaginary quadratic fields are fractionally dense in $\mathbb{R}_{>0}$ and $\mathbb{C}$ respectively.