Continued Fraction Expansion via Gauss Map

Theorem

Let $T : \closedint 0 1 \to \closedint 0 1$ be the Gauss map.

Let $x \in \closedint 0 1 \setminus \Q$.

Then $x$ has the simple infinite continued fraction:

\(\ds x\)

\(=\)

\(\ds \sqbrk {0; \map {a_1} x, \map {a_2} x, \ldots}\)

\(\ds \)

\(=\)

\(\ds 0 + \cfrac 1 {\map {a_1} x + \cfrac 1 {\map {a_2} x + \cfrac 1 \ddots} }\)

where:

$\map {a_n} x := \floor {\dfrac 1 {\map {T^{n - 1} } x} }$

$\floor \cdot$ denotes the floor.

Proof

For $x \in \closedint 0 1 \setminus \Q$, we have:

$\map {a_1} x = \floor {\dfrac 1 x}$

and:

$\forall n \in \N_{>0} : \map {a_n} x = \map {a_1} { T^{n-1} x }$

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