Skip to main content
Log in

On best two-dimensional Dirichlet-approximations and their algorithmic calculation

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. Barbour, J.M.: Music and ternary continued fractions. Amer. Math. Monthly55, 545–551 (1948)

    Google Scholar 

  2. Bernstein, L.: The Jacobi-Perron algorithm. Its theory and application. Lecture Notes in Mathematics. Berlin, Heidelberg, New York: Springer 1971

    Google Scholar 

  3. Brun, V.: Mehrdimensionale Algorithmen, welche die Eulersche Kettenbruchentwicklung der Zahlen verallgemeinern. Sammelband zu Ehren des 250. Geburtstages Leonhard Eulers. Berlin: Akademie 1959

    Google Scholar 

  4. Brun, V.: Music and Euclidean algorithm. Nordisk Mat. Tidskr.9, 29–36 (1961)

    Google Scholar 

  5. Cassels, J.W.S.: An introduction to Diophantine approximation. Cambridge: University Press 1957

    Google Scholar 

  6. Cusick, T.W.: The Szekeres multidimensional continued fraction. Math. Comput.31, 280–317 (1977)

    Google Scholar 

  7. Jacobi, C.G.J.: Allgemeine Theorie der kettenbruchähnlichen Algorithmen. J. Reine Angew. Math.69, 417–436 (1868)

    Google Scholar 

  8. Jurkat, W., Kratz, W., Peyerimhoff, A.: Explicit representations of Dirichlet approximations. Math. Ann.228, 11–25 (1977)

    Google Scholar 

  9. Koksma, J.F.: Diophantische Approximationen. New York: Chelsea 1936

    Google Scholar 

  10. Lekkerkerker, C.G.: Geometry of numbers. Amsterdam: Wolters-Noordhoff 1969

    Google Scholar 

  11. Mahler, K.: Über die Annäherung algebraischer Zahlen durch periodische Algorithmen. Acta Math.68, 109–144 (1937)

    Google Scholar 

  12. Minkowski, H.: Gesammelte Abhandlungen I, 271–277, 278–292, 293–315, 357–370. New York: Chelsea 1967

    Google Scholar 

  13. Perron, O.: Die Lehre von den Kettenbrüchen, I, II. Stuttgart: Teubner 1957

    Google Scholar 

  14. Perron, O.: Grundlagen für eine Theorie des Jacobischen Kettenbruchalgorithmus. Math. Ann.64, 1–76 (1907)

    Google Scholar 

  15. Pipping, N.: Ein Kriterium für die reellen algebraischen Zahlen auf eine direkte Verallgemeinerung des Euklidischen Algorithmus gegründet. Acta Acad. Abo. Ser. B.1 (1921)

  16. Pipping, N.: Approximation zweier reeller Zahlen durch rationale Zahlen mit gemeinsamen Nenner. Acta Acad. Abo. Ser. B.27, 1–17 (1957)

    Google Scholar 

  17. Selmer, E.S.: Continued fractions in several dimensions. Nordisk Mat. Tidskr.9, 37–43 (1961)

    Google Scholar 

  18. Szekeres, G.: Multidimensional continued fractions. Ann. Univ. Sci. Budapest Eötvös Sect. Math.13, 113–140 (1970)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

The work of the first author was supported in part by the National Science Foundation

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jurkat, W., Kratz, W. & Peyerimhoff, A. On best two-dimensional Dirichlet-approximations and their algorithmic calculation. Math. Ann. 244, 1–32 (1979). https://doi.org/10.1007/BF01420334

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01420334

Keywords

Navigation