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Hi, I am a student researcher trying to prove that all irrationals within the Cantor set are transcendental. This is grounded, intuitively, in Cantor set members' being non-normal; since algebraic numbers are widely believed to be normal, this implies the transcendentality of the irrationals in the Cantor set. Now, I am at a dead end in trying to prove this, and I would appreciate any pointers (I will give you full credit in whatever end product comes out of this).

I tried recasting the problem in the following way: if I can prove that any irrational algebraic number must have a $1$ in its ternary expansion, then the result I'm after follows. With this, I realized that if an irrational algebraic number has at least one $1$ in its ternary expansion, then it must have infinitely many of them (because irrational algebraics are a field, and otherwise I could add an appropriate irrational algebraic number and get another irrational algebraic without any $1$'s in their ternary expansions -- a contradiction). But I can't really get anywhere past this (tried playing with the field properties, too). Mostly, I just lack/do not know of the tools used in this kind of problem.

Any help very much appreciated!

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    $\begingroup$ This is a very open-ended question. If this is an open problem and due to be part of your thesis then I imagine many MO participants would feel quite uncomfortable offering their help, 'full credit' given or not. Perhaps you could clarify your status. Are you studying for a PhD? What does your advisor say? $\endgroup$
    – HJRW
    Nov 28, 2012 at 11:28
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    $\begingroup$ @HW: I'm a bachelor's student and this would be for my undergrad thesis. The topic of the thesis is to simply explore the cantor set and provide a writeup of what I learned from a number of papers. This question I'm interested in looking at on my own and was planning to include my work on it in the paper. I was hoping for 'perhaps you could look at such and such property' kind of pointers, but if that's too much to ask for I understand. $\endgroup$
    – CantorSet
    Nov 28, 2012 at 11:52
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    $\begingroup$ I don't work in this area of mathematics, but my impression is that this sort of problem would, at a realistic minimum, take years of work. Lasse's report seems to support that impression. It's a good question, but too ambitious to realistically hope to get anywhere on in your time frame, IMO. $\endgroup$
    – Todd Trimble ♦
    Nov 28, 2012 at 12:10

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This question was asked by Mahler ("Some suggestions for further research", Bull. Austral. Math. Soc. 29 (1984), no. 1, 101–108).

See Adamczewski, Bugeaud, "On the decimal expansion of algebraic numbers" (2005) for some things that are known. I have confirmed with a colleague that this is still very much a (widely) open problem. It does not appear that this problem would be appropriate for undergraduate research.

Of course this is not to say that learning about it would not be beneficial, or that there could not be related research problems that are more tractable. I suggest that you should seek the guidance of an experienced researcher in the field.

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  • $\begingroup$ Thanks, this is the kind of suggestions of I was looking for! I will dabble in the matter and probably just summarize what people have done in their attempts. $\endgroup$
    – CantorSet
    Nov 28, 2012 at 12:13
  • $\begingroup$ You may also be interested in Bugeaud's article, "Diophantine approximation and Cantor sets", Mathematische Annalen, Volume 341, Number 3, July 2008 , pp. 677-684(8). This solves the other problem that Mahler states regarding the Cantor set. $\endgroup$
    – Lasse Rempe
    Nov 28, 2012 at 12:17
  • $\begingroup$ Is this problem still open, nearly 5 years later? $\endgroup$
    – user21820
    Oct 16, 2017 at 15:08
  • $\begingroup$ As far as I am aware it remains open. $\endgroup$
    – Lasse Rempe
    Nov 22, 2017 at 11:11

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