Some Existence Results on Cantor Sets

TL;DR

This paper proves the existence of two distinct Cantor sets within the sets of Liouville and Diophantine numbers and provides a condition for the existence of Cantor sets in subsets of the real line.

Contribution

It establishes the existence of specific Cantor sets in special number sets and characterizes when Cantor sets can exist within subsets of the real line.

Findings

Existence of Cantor sets in Liouville and Diophantine numbers

Necessary and sufficient condition for Cantor sets in subsets of the real line

Construction methods for such Cantor sets

Abstract

The existence of two different Cantor sets, one of them contained in the set of Liouville numbers and the other one inside the set of Diophantine numbers, is proved. Finally, a necessary and sufficient condition for the existence of a Cantor set contained in a subset of the real line is given.