A number is rational iff its binary expansion is repeating. My main question is: What restrictions does the property of beeing an irrational, but algebraic number place on its binary expansion?
Can we prove that for every irrational algebraic number the limiting ratio of the number of 1's to the number of 0's in the first $n$ digits of its binary expansion, exists and is equal to $1/2$?
Is there an algebraic number such that every finite sequence of 1's and 0's occurs in its binary expansion, infinite times?