Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-20T06:26:57.843Z Has data issue: false hasContentIssue false
  • English
  • Français

Almost everywhere exponential convergence of the modified Jacobi—Perron algorithm

Published online by Cambridge University Press:  19 September 2008

S. Ito ,
M. Keane  and
M. Ohtsuki
S. Ito
Affiliation:
Department of Mathematics, Tsuda College, Tsuda-machi, Kodaira, Tokyo, Japan
M. Keane
Affiliation:
Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands
M. Ohtsuki
Affiliation:
Department of Mathematics, Tsuda College, Tsuda-machi, Kodaira, Tokyo, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that there exists a constant δ > 0 such that for almost every pair of numbers α and β there exists n0= n0(α,β) such that for any n ≥ n0

where the integers pn,qn, rn are provided by the modified Jacobi-Perron algorithm.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 13 , Issue 2 , June 1993 , pp. 319 - 334
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baldwin, P. R.. A multidimensional continued fraction and some of its statistical properties. J. Stat. Phys. 66 (1992), 1463–1505.CrossRefGoogle Scholar
[2]Baldwin, P. R.. A convergence exponent for multidimensional continued-fraction algorithms. J. Slat. Phys. 66 (1992), 1507–1526.Google Scholar
[3]Cassels, J. W. S.. An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge, 1957.Google Scholar
[4]Katznelson, Y. & Weiss, B.. A simple proof of some ergodic theorems. Israel J. Math. 42 (1982), 291–296.Google Scholar
[5]Kingman, J. F. C.. The ergodic theory of subaddive stochastic processes. J. Royal Stal. Soc. B 30 (1968), 499–510.Google Scholar
[6]Krengel, U.. Ergodic Theorems. DeGruyter Studies in Mathematics 6. DeGruyter, Berlin-New York, 1985.CrossRefGoogle Scholar
[7]Podsypanin, E. V.. A generalization of the continued fraction algorithm that is related to the Viggo Brun algorithm (Russian). Studies in Number Theory (LOMI), 4. Zap. Naucn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 67 (1977), 184–194.Google Scholar
[8]Schweiger, F.. The metrical theory of the Jacobi-Perron algorithm. Springer Lecture Notes in Mathematics 334. Springer, Berlin, 1973.Google Scholar
[9]Schweiger, F.. A modified Jacobi-Perron algorithm with explicitly given invariant measure. Ergodic Theory. Springer Lecture Notes in Mathematics. Denker, M. and Jacobs, K., eds, Springer, Berlin, 1978, 199–202.Google Scholar
[10]Schweiger, F.. Invariant measures for maps of continued fraction type. J. Number Theory 39 (1991), 162–174.CrossRefGoogle Scholar