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Arithmetic distribution of denominators of convergents of continued fractions. (Distributions arithmétiques des dénominateurs de convergents de fractions continues.) (French) Zbl 0655.10045
Let $$x=[a_ 1,a_ 2,a_ 3,...]$$ denote the continued fraction expansion. As usual denote by $$p_ n(x)$$ and $$q_ n(x)$$ the numerator and denominator of the $$n$$th convergent. Furthermore we put $$Tx=(1/x) mod 1=[a_ 2,a_ 3,a_ 3,...]$$. It is known that $$d\mu /d\lambda =((1+x) \log 2)^{-1}$$ is the density of the invariant measure.
The first main theorem is the following: If $$x$$ is a quadratic irrational number then the sequences $$(\log p_ n(x))$$ and $$(\log q_ n(x))$$ are uniformly distributed mod 1. From this it follows that the sequences $$(p_ n(x))$$ and $$(q_ n(x))$$ obey Benford’s law.
The second main theorem uses the group of $$2\times 2$$-matrices $$G(m)$$ the entries of which are integers modulo $$m$$ and with determinant $$\pm 1$$. We denote its Haar measure by $$h(m)$$. If $$g=(\alpha,\beta,\gamma,\delta)\in G(m)$$ then we put $$g.a=(\beta,\alpha +\beta a,\delta,\gamma +\delta a).$$ Then it is shown that the map $$S$$ on $$[0,1[\times G(m)$$ into itself defined as $$S(x,g)=(Tx, g.a_ 1(x))$$ is ergodic with respect to $$\mu$$ $$\otimes h(m).$$
From this theorem several results on the distribution of $$q_ n(x) \bmod m$$ can be deduced. Related results have also been obtained by P. Szüsz [Acta Arith. 8, 225–241 (1963; Zbl 0123.04602)].
Reviewer: F.Schweiger

MSC:
 11K50 Metric theory of continued fractions 11K06 General theory of distribution modulo $$1$$ 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc. 28D05 Measure-preserving transformations