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arXiv · 2506.13103

Descriptions of Cantor Sets: A Set-Theoretic Survey and Open Problems

Mohsen Soltanifar
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Abstract

This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date taxonomy, we review the Borel hierarchy and four hierarchically ordered representations-general, nested, iterated-function-system (IFS), and q-ary expansion-presented from the most general to the most specific set-theoretic description of deterministic Cantor sets. We then present explicit and recursive descriptions for two thin families of measure-zero Cantor sets and an augmented "tick" family of positive measure, respectively, showing that the classical middle-third set lies in the intersection of all three families of after-mentioned Cantor sets. The survey closes by isolating several open problems in four directions, aiming to provide mathematicians with a coherent platform for further descriptive set-theoretic investigations into Cantor-type sets on the real line.

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BibTeXRIS

Mohsen Soltanifar. 2025-06-16. Descriptions of Cantor Sets: A Set-Theoretic Survey and Open Problems. https://doi.org/10.69882/adba.chf.2026071

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