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Multidimensional continued fractions and stable laws. (Fractions continues multidimensionnelles et lois stables.) (French) Zbl 0857.11035
The Jacobi-Perron algorithm is a multidimensional version of the classical continued fraction expansion. Starting from an \(x\in [0,1]^d\) one computes \(\overline x = (x_2/x_1, x_3/x_1, \dots, 1/x_1)\), and \(a(x) = [\overline x]\), \(Tx = \{\overline x\}\), where \([ ]\) and \(\{ \}\) denote the integer and fractional part, respectively. The first approximant of \(x\) is then defined as \((1/a_d,a_1/a_d, \dots, a_{d-1}/a_d)\), and the rest is \(Tx\). One has \(x=(1/(a_d + (Tx)_d)\), \((a_1+(Tx)_1)/(a_d+(Tx)_d), \dots, (a_{d-1}+(Tx)_{d-1})/(a_d+(Tx)_d))\). Continuing this procedure for all \(T^kx\) as in the one dimensional case, one gets simultaneous rational approximations of the numbers \(x_1,\dots,x_d\) of the form \(r_n=(p^1_n/q_n, \dots, p^d_n/q_n)\). This algorithm relies on the iteration of the transformation \(T\). It is well known (and goes back essentially to Gauss) that in the one dimensional case the measure \(T^n \lambda\), where \(\lambda\) is the Lebesgue measure on \([0,1]\), converges towards the measure \({1 \over \log 2} {dx \over 1+x}\) which is the unique probability measure invariant under the transformation \(T\), and absolutely continuous with respect to Lebesgue measure. This fact is extended in the paper to the multidimensional Jacobi-Perron algorithm, the density being now a piecewise analytic function on \([0,1]\).
Next the author considers the problem of the size of the integer vectors \(a(T^nx)\), which are the coefficients in the continued fraction expansion of \(x\), as \(n\to\infty\). It is shown that for all multi-indices \(\alpha=(\alpha_1, \dots, \alpha_d)\) where the \(\alpha_j\) are positive real numbers, and \(\overline \alpha=\sum_j \alpha_j>1/2\), the random variables \(1/n^{\overline \alpha} \sum^{n-1}_{k=0} (-1)^k a(T^k x)^\alpha\) (where \(x\) is chosen with Lebesgue measure) converge in distribution towards a multidimensional symmetric stable distribution of index \(1/ \overline \alpha\). This extends a result due to Paul Lévy in the one-dimensional case.
Reviewer: Ph.Biane (Paris)

MSC:
11J70 Continued fractions and generalizations
37A99 Ergodic theory
60F05 Central limit and other weak theorems
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