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Goles And S. Martinez Eds. Arnoux; V. Ferenczi, C. Mauduit, A. Siegel Eds. Arnoux; S. Ito Pisot substitutions and Rauzy fractals , Bull. Simon Stevin , Tome 8 , pp. Ferenczi; P. Hubert Trajectories of rotations , Acta Arith. Ito; Y.
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- The Metrical Theory of Jacobi-Perron Algorithm?
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Sano Higher dimensional extensions of substitutions and their dual maps , J. Arnoux; G. France , Tome , pp. Berstel; Dassow, Rozenberg, Salomaa, Eds.
Berstel; P. Vuillon Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences , Discrete Math. Nombres Bordeaux , Tome 12 , pp. Vuillon Palindromes and two-dimensional Sturmian sequences , J. Broise Fractions continues multidimensionnelles et lois stables , Bull. Broise-Alamichei; Y. Abstract: Multidimensional continued fractions MCFs were introduced by Jacobi and Perron in order to generalize the classical continued fractions. In this paper, we propose an introductive fundamental study about MCFs in the field of the -adic numbers.
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First, we introduce them from a formal point of view, i. In particular, we derive some sufficient conditions for their convergence and we prove that convergent MCFs always strongly converge in contrary to the real case where strong convergence is not always guaranteed. Then, we focus on a specific algorithm that, starting from an -tuple of numbers in odd , produces the partial quotients of the corresponding MCF. We see that this algorithm is derived from a generalized -adic Euclidean algorithm and we prove that it always terminates in a finite number of steps when it processes rational numbers.
References [Enhancements On Off] What's this? Pura Appl.
MR  A. Bryuno and V. Parusnikov , Comparison of various generalizations of continued fractions , Mat.
Zametki 61 , no. Notes 61 , no. I , Demonstratio Math. MR  Jerzy Browkin , Continued fractions in local fields. II , Math. Number Theory 10 , no.
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A-B , AA French. Number Theory , 1— One of the first theorems in metric number theory is Borel's theorem E. Borel, : When written in an arbitrary fixed integer base , almost-all in the sense of Lebesgue measure real numbers in the interval are normal cf. Normal number. In an equivalent formulation, this theorem asserts that the fractional parts , are uniformly distributed in the interval.
Borel's theorem has been generalized and extended by many mathematicians. It turned out to be fruitful to take the view that the "digits" in the base expansion of in the -ary system are independent random variables. Founded explicitly or implicitly on this situation, and by applying methods developed in probability theory for finding the asymptotic distributions of sums of independent or weakly-dependent random variables, the basic questions regarding the distribution of the "digits" , and arbitrary groups of "digits" , in the -ary expansion of numbers "randomly" chosen from , were solved.
For example, putting. If is any number from , and denotes the number of for which for a given , then. Weyl proved that if , is an arbitrary increasing sequence of natural numbers, then for almost-all the fractional parts are uniformly distributed in the interval. On the assumption that the are the values of a function defined on the infinite interval and satisfying special analytic conditions, this theorem has a more precise version concerning the "quality" of the uniform distribution.
Koksma  proved a general theorem on the distribution of the fractional parts cf.
Distribution modulo one of a function in two variables , where is a real number taking almost-all values from the interval , and. For example, the fractional parts , for almost-all , are uniformly distributed in the interval. Apart from questions connected with Borel normal numbers, one of the basic objects in metric number theory, from the start of its development, was the metric theory of continued fractions. Let be a real number in the interval , let be its continued fraction expansion, let , and let be the denominator of the -th convergent. Khinchin established that for almost-all , as ,.
In addition, Khinchin  used his results on the metric properties of continued fractions to prove a theorem on the approximation of numbers by rationals. Let be a positive continuous function of a positive argument such that is a non-increasing function. Then the inequality. This theorem has been transcended and generalized from various points of view. It became the starting point of an intensive development of the metric theory of Diophantine approximation.
A necessary and sufficient condition under which for almost-all an infinite number of are contained in an arbitrary sequence is. Under the same condition, for almost-all the inequality.
These results are close to the conjecture : If is an arbitrary sequence of integers and are arbitrary, then the inequality. Kuz'min proved that for any the measure of the set of 's for which , is equal to. The asymptotic relation. Gauss, but he did not publish it, and he mentions in one of his letters to P.
The Metrical Theory of Jacobi-Perron Algorithm by Fritz Schweiger - ozuduvocelak.ml
Laplace that it would be very desirable to estimate the difference. Kuz'min's method was the source of many other metric theorems on continued fractions. The modern treatment of metric questions connected with Borel normal numbers and the theory of continued fractions uses ideas of ergodic theory. This is based on the fact that the mappings and of the interval into itself, which are closely connected with the -ary fraction and the continued fraction respectively, preserve a measure and are ergodic: the first preserves the Lebesgue measure, the second the measure defined for every measurable set in by.
From this point of view the Gauss—Kuz'min theorem, without estimates on the remainder, follows immediately from Birkhoff's individual ergodic theorem. Arguments from ergodic theory turn out to be useful even in order to obtain estimates of the remainder in certain limit theorems. For example, the results of Khinchin admit the improvement see . The ideas of ergodic theory are useful in many other problems of the metric theory of numbers linear Diophantine equations, the distribution of values of matrix exponential functions, the Jacobi—Perron algorithm, and others.
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