Ein Gauss-Kusmin-Levy-Satz f�r Kettenbr�che nach n�chsten Ganzen

Rieger, G. J.

doi:10.1007/bf01168885

Cited by 11 publications

(9 citation statements)

References 2 publications

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“…the aforementioned diagonal continued fraction (DCF), see [K1]); In order to apply the OCF-algorithm "one needs to know where one has been". It is exactly this aspect of the OCF which makes it very difficult − if not impossible − to obtain a Gauss-Kusmin theorem for the OCF in the same vein as those obtained for the NICF, SCF or for the RCF (it should be noticed that the approach from [Wir] and [Ba] cannot be used for the NICF or the SCF, see also [Rie1], p. 444).…”

Section: Karma Dajani and Cor Kraaikampmentioning

confidence: 99%

“…For further reference we will mention here a slightly modified version of Rieger's 1978 version of the Gauss-Kusmin theorem for the SCF; see also in [Rie1] the proof of Satz 2 and (7.1).…”

Section: A Two Dimensional Gauss-kusmin Theoremmentioning

confidence: 99%

“…1. A similar theorem can be formulated for the NICF; see [Rie1], Satz 2, and also [Roc]. In this paper we choose to work with the SCF instead of the NICF only because the natural extension of the SCF is "slightly nicer" than the one for the NICF (see also [Na], [K1]); one simply needs to discern fewer cases in the proofs of the various results in case one uses the SCF.…”

Section: Theorem 21 For Every Borel Set E ⊂ X G One Hasmentioning

confidence: 99%

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A Gauss-Kusmin theorem for optimal continued fractions

Dajani

Kraaikamp

1999

Trans. Amer. Math. Soc.

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Abstract. A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal S-expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal S-expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal S-expansion.

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Section: Karma Dajani and Cor Kraaikampmentioning

confidence: 99%

“…For further reference we will mention here a slightly modified version of Rieger's 1978 version of the Gauss-Kusmin theorem for the SCF; see also in [Rie1] the proof of Satz 2 and (7.1).…”

Section: A Two Dimensional Gauss-kusmin Theoremmentioning

confidence: 99%

Section: Theorem 21 For Every Borel Set E ⊂ X G One Hasmentioning

confidence: 99%

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A Gauss-Kusmin theorem for optimal continued fractions

Dajani

Kraaikamp

1999

Trans. Amer. Math. Soc.

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“…The case where [·] is the floor function and S = [0, 1] is the most familiar; see [4] for a brief account. The case where [·] rounds to the nearest integer and S = [−1/2, 1/2] is qualitatively similar; see [11].…”

Section: Introductionmentioning

confidence: 91%

A multidimensional continued fraction based on a high-order recurrence relation

Tourigny

Smart

2007

Math. Comp.

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Abstract. The paper describes and studies an iterative algorithm for finding small values of a set of linear forms over vectors of integers. The algorithm uses a linear recurrence relation to generate a vector sequence, the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values, subject to the requirement that the integers should not become too large. The problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one-parameter family of lattices; the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero. The paper discusses two rounding procedures to solve the underlying inhomogeneous Diophantine approximation problem: the first, which we call "naive rounding" leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties; in particular, when it is applied to the familiar problem of simultaneous rational approximation, the algorithm reduces to the classical Jacobi-Perron algorithm. The second rounding procedure is Babai's nearest-plane procedure. We compare the two rounding procedures numerically; our experiments suggest that the multidimensional continued fraction corresponding to nearest-plane rounding converges at an optimal asymptotic rate.

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“…The problems empirically studied by the author [1] have earlier been completely solved by Rieger [2], [3] and Rockett [4]. See also [5].…”

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confidence: 98%

On the metric theory of nearest integer continued fractions

Riesel¹

1988

BIT

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The problems empirically studied by the author [1] respectively, where G = (1+x/5)/2. The numerical valu.es given for re(t) are quite accurate; the largest error is 0.64.10-5, occurring for t = 0.14. The values for m'(t) differ by up to 3 units in the last decimal, except the value for m'(0), which differs by 4 units. The exact value of the constant In K' in (38) and (41) is u2/121n G = 1.709...The ratio 70 ~ at the end ofsubsection 17 is In G/ln 2 = 0.6942... H. Riese••

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Ein Gauss-Kusmin-Levy-Satz f�r Kettenbr�che nach n�chsten Ganzen

Cited by 11 publications

References 2 publications

A Gauss-Kusmin theorem for optimal continued fractions

A Gauss-Kusmin theorem for optimal continued fractions

A multidimensional continued fraction based on a high-order recurrence relation

On the metric theory of nearest integer continued fractions

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