# Transcendentality of all irrationals in the Cantor set

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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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? –  HJRW Nov 28 '12 at 11:28
@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. –  CantorSet Nov 28 '12 at 11:52
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. –  Todd Trimble Nov 28 '12 at 12:10