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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

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