Borel conjectured that all irrational numbers are normal in any integral base α. For each positive number ξ and integer α greater than 1, ξ is normal in base α if and only if the sequence is uniformly distributed modulo 1. In this paper we survey not only the digit of algebraic irrational numbers in integral base but also the fractional parts of geometric progressions whose common ratios are algebraic numbers greater than 1. In our main results, we give new lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.
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© 2010 American Institute of Physics.
2010
American Institute of Physics
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