Cantor set and Base 3 expansion

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SUMMARY

The discussion centers on the ternary expansion of numbers and its relationship to the Cantor set. Specifically, a number belongs to the Cantor set formed after the k-th iteration if and only if each digit in its base 3 expansion is either 0 or 2, excluding 1. This property is crucial for understanding the Cantor set's structure and can be proven through the analysis of base 3 representations. Participants express gratitude for clarifying this concept and the availability of resources.

PREREQUISITES
  • Understanding of Cantor set iterations
  • Familiarity with base 3 (ternary) number systems
  • Knowledge of decimal and ternary expansions
  • Basic proof techniques in mathematics
NEXT STEPS
  • Study the properties of the Cantor set and its iterations
  • Learn about base 3 expansion and its applications
  • Explore mathematical proofs related to set theory
  • Investigate the relationship between Cantor sets and fractals
USEFUL FOR

Mathematicians, students studying set theory, and anyone interested in fractals and number theory will benefit from this discussion.

barksdalemc
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How do I do a ternary expansion of numbers, and prove that if a number is part of the 2^k iteration of the cantor set if and only if each decimal expansion position is either a two or zero? If you guys can give me a hint, I would love to go from there.
 
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I think you have that wrong. A number is in the set formed after the kth iteration of the process of removing middle thirds used to create the cantor set iff the kth digit in its base 3 expansion is not 1 (more specifically (to handle endpoints like 1/3=0.1=0.0222...), if it can be written in such a form). Since this is true, maybe it will be easier to prove.
 
Yep

You are right. Thanks a lot. I didnt realize this resource was availabe and hopefully will be able to contribute both ways.
 

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