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!



Transcendentality of all irrationals in the Cantor set
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.


