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9

Continued fraction expansions and β-expansions

Beside b-ary expansions, there are many classical ways to represent a real number, including by means of its continued fraction expansion, or of its β-expansion. In the first section, we define the notion of normal continued fraction and construct explicitly a real number having a normal continued fraction. In the next section, we present several results on the continued fraction expansion of algebraic numbers. Section 9.3 is devoted to a short survey on β-expansions.

9.1  Normal continued fractions

The sequence of partial quotients of an irrational real number ξ in (0, 1) can be obtained by iterations of the Gauss map defined by (0) = 0 and (x) = {1/x} for x ∈ (0, 1). Namely, if [0; a1, a2

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