Cantor Set


by Office_Shredder
Tags: cantor
Office_Shredder
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#1
Dec9-07, 04:29 PM
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1. The problem statement, all variables and given/known data
If C is the cantor set, prove that [tex]{x \in R | x=t_1+t_2, t_1, t_2 \in C} = [0,2] [/tex]. In english if that wasn't clear, show the set of all numbers that are the sum of two cantor set elements is precisely [0,2]


2. Relevant equations
The Cantor set of course being constructed on the interval [0,1]


3. The attempt at a solution
I know that the elements of the Cantor set are precisely those that can be written in base 3 with only 0's and 2's, so I thought maybe for a generic base 3 element, I could construct two cantor set elements that sum to it. This didn't work because I needed to worry about carrying over from infinitely far away (since when constructing the two cantor set elements, I obviously need to start at the first decimal place, but addition 'starts' at the infinitieth or whatever you want to call it) so that didn't pan through so well. What's the best way to start this? It's obvious the set of sum of two cantor elements is a subset of [0,2], but any attempt to go the other way just ends in failure.
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Dick
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#2
Dec9-07, 05:01 PM
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You seem to be saying that you wouldn't have any trouble constructing a such a sum if the number of digits were finite. Isn't that good enough? If you have a_n+b_n approaching some number c in [0,2] with a_n and b_n all in the Cantor set then the sequences a_n and b_n have a cluster point in the Cantor set, since it's compact.
Office_Shredder
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#3
Dec9-07, 05:28 PM
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Quote Quote by Dick View Post
You seem to be saying that you wouldn't have any trouble constructing a such a sum if the number of digits were finite. Isn't that good enough? If you have a_n+b_n approaching some number c in [0,2] with a_n and b_n all in the Cantor set then the sequences a_n and b_n have a cluster point in the Cantor set, since it's compact.
Ah, of course. Thanks


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