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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?

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Being algebraic does place a restriction on how a number can be approximated by rational numbers: numbers that are closely approximated by rationals are transcendental. See Liouville number.

For the last question, the notion is captured in normal number. Wikipedia says that "it is widely believed that $\sqrt 2$ is normal, but a proof remains elusive."

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    Indeed, it is widely believed that every real algebraic irrational is normal in every base, b$u$t nothing whatsoe$v$er has been proved along those lines.2012-02-14