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Math Help - Cantor ternary set

  1. #1
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    Cantor ternary set

    Are there any irrational algebraic numbers in the Cantor ternary set?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bobbyk
    Are there any irrational algebraic numbers in the Cantor ternary set?
    Interesting question. I don't know the answer (yet - I hope), but as web
    searches have turned up nothing relevant its either trival or the proof of
    (non-)existence of such is obscure.

    My question is: In what context has this question been asked? Is this
    homework, research or something else?

    RonL
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  3. #3
    Super Member Rebesques's Avatar
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    Tried a calculation, and saw that it can contain no n-th roots. This is too harsh a condition, so I bet it doesn't contain any algebraic irrationals at all. Don't have a complete proof, though.
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  4. #4
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    Quote Originally Posted by CaptainBlack
    Interesting question. I don't know the answer (yet - I hope), but as web
    searches have turned up nothing relevant its either trival or the proof of
    (non-)existence of such is obscure.

    My question is: In what context has this question been asked? Is this
    homework, research or something else?

    RonL
    No, it's not homework or research. It's just something I've wondered about
    for many years and have asked a number of mathematicians with no answers.

    bobbyk
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by bobbyk
    Are there any irrational algebraic numbers in the Cantor ternary set?
    The Wikipedia article on Normal Number* has something to say about
    this:

    "David H. Bailey and Richard E. Crandall conjectured in 2001 that every
    irrational algebraic number is normal; while no counterexamples are known,
    not a single irrational algebraic number has ever been proven normal in any
    base"

    Now if the Bailey-Crandall conjecture is true then Cantors ternary set would
    contain no irrational algebraic numbers, as in base-3 the ternary set contains
    no base-3 normal numbers.

    (Note the Wikipedia article refers to normality but the Bailey-Crandall paper
    refers to absolute normality which means normality in all bases)

    RonL

    * A number N is normal to base-b if the base-b expansion contains every
    sequence of digits of length m with relative frequency b^{-m}. Which
    essentialy means all digits, and strings of digits appear in the expansion of N
    with the frequency that they would be expected in a random string.
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  6. #6
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    Thanks for noticing this conjecture! I hadn't heard of it. So my question is
    still unanswered, but is probably in the negative, since no counterexamples
    have been found. I was going to ask Ulam about this, but he died before I
    could get around to it
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  7. #7
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    What is a ternary set?
    I was not able to find it on Wikipedia however it did have a Cantor set.
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  8. #8
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker
    What is a ternary set?
    I was not able to find it on Wikipedia however it did have a Cantor set.
    Its the same thing

    RonL
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by bobbyk
    Thanks for noticing this conjecture! I hadn't heard of it. So my question is
    still unanswered, but is probably in the negative, since no counterexamples
    have been found. I was going to ask Ulam about this, but he died before I
    could get around to it
    I'm impressed by the name dropping, how did you come to know Ulam?

    RonL
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  10. #10
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    Ulam

    Sorry about the name dropping! I didn't know Ulam. I'm not even a mathematician. He was just one of my heroes.
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