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!