On the complexity of the binary expansions of algebraic irrational numbers (survey)

Borel conjectured that all irrational numbers are normal in any integral base alpha. For each positive number xi and integer alpha greater than 1, xi is normal in base alpha if and only if the sequence xi alpha(n) (n = 0,1,...) 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.