Estimation of the constant of the best simultaneous diophanite approximations for $n=5$ and $n=6$

Yu. A. Basalov

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 66-81.

Résumé

This paper is devoted to the problem of estimating from below the constant of the best Diophantine approximations for $ n $ real numbers. This problem is a special case of the more general problem of approximating $ n $ real linear forms and has its rich history, ascending to P. G. Dirichlet. A significant contribution at an early stage of research was made by A. Hurwitz using the apparatus of continued fractions and F. Furtwängler, using the apparatus of linear algebra. In the mid-twentieth century, H. Davenport found a fundamental connection between the value of the constant constant of the best Diophantine approximations and critical determinant of a special type of star body. Later J. W. S. Cassels switched from directly calculating the critical determinant to estimating its value by calculating the largest value of $V_{n,s}$ – the volume of a parallelepiped centered at the origin of coordinates with certain properties. This approach allowed us to obtain estimates from below of the constant of the best joint Diophantine approximations for $ n = 2, 3, 4 $ (see the works of J. W. S. Cassels, T. Cusick, S. Krass). In this paper, based on the approach described above, estimates for $n=5$ and $n=6$ are obtained. The idea of building estimates is different from the work of T. Cusick. Using numerical experiments we approximate and then obtaine exact values of the estimates of $V_{n,s}$. The proof of these estimates is rather cumbersome and is primarily a technical difficulty. Another different of the given estimates is the ability to generalize them to any dimension. In the process of numerical experiments was also obtained interesting information about the structure of the $V_{n,s}$ values. These results agree quite well with the results obtained in the works of S. Krass. The question of the structure of the values of $V_{n,s}$ for large dimensions has been scantily explored and can be of considerable interest both from the point of the geometry of numbers and from the point of the theory of Diophantine approximations.

Détail

Keywords: best joint Diophantine approximations, geometry of numbers, star bodies, critical determinants.

Bibliographie

[1] Basalov Yu. A., On estimating the constant of simultaneous Diophantine approximation, arXiv: (data obrascheniya: 09.04.2019) 1804.05385

[2] Cassels J. W. S., “Simultaneous Diophantine approximation”, J. London Math. Soc., 30 (1955), 119–121 | MR | Zbl

[3] Cusick T. W., “Estimates for Diophantine approximation constants”, Journal of Number Theory, 12:4 (1980), 543–556 | MR | Zbl

[4] Davenport H., “On a theorem of Furtwängler”, J. London Math. Soc., 30 (1955), 186–195 | MR | Zbl

[5] Finch S. R., Mathematical Constants, Encyclopedia of Mathematics and its Applications, 94, Cambridge University Press, 2003 | MR | Zbl

[6] Furtwängler H., “Über die simulatene Approximation von Irrationalzahlen. I”, Math. Ann., 96 (1927), 169–175 | MR

[7] Furtwängler H., “Über die simulatene Approximation von Irrationalzahlen. II”, Math. Ann., 99 (1928), 71–83 | MR | Zbl

[8] Hurwitz A., “Über die angenaherte Darstellung der Irrationalzahlen durch rationale Briiche”, Math. Ann., 39 (1891), 279–284 | MR

[9] Krass S., “Estimates for $n$-dimensional Diophantine approximation constants for $ n \geq 4 $”, J. Number Theory, 20:2 (1985), 172–176 | MR | Zbl

[10] Krass S., “The $N$-dimensional diophantine approximation constants”, Australian Mathematical Society, 32:2 (1985), 313–316 | MR | Zbl

[11] Mack J. M., “Simultaneous Diophantine approximation”, J. Austral. Math. Soc. A, 24 (1977), 266–285 | MR | Zbl

[12] Nowak W. G., “A note on simultaneous Diophantine approximation”, Manuscr. Math., 36 (1981), 33–46 | MR | Zbl

[13] Nowak W. G., “A remark concerning the s-dimensional simultaneous Diophantine approximation constants”, Graz. Math. Ber., 318 (1993), 105–110 | MR | Zbl

[14] Nowak W. G., “Lower bounds for simultaneous Diophantine approximation constants”, Comm. Math., 22:1 (2014), 71–76 | MR | Zbl

[15] Nowak W. G., “Simultaneous Diophantine approximation: Searching for analogues of Hurwitz's theorem”, Essays in mathematics and its applications, eds. T. M. Rassias, P. M. Pardalos, Springer, Switzerland, 2016, 181–197 | MR | Zbl

[16] Yu. A. Basalov, “On the history of estimates of the constant of the best joint Diophantine approximations”, Chebyshevskii sbornik, 19:2 (2018), 388–405 | DOI

[17] J. W. S. Cassels, An Introduction to the Geometry of Numbers, Mir, 1965 | MR

[18] N. G. Moshchevitin, “To the Blichfeldt-Mullender-Spohn theorem on simultaneous approximations”, Proceedings of the Steklov Mathematical Institute, 239 (2002), 268–274

[19] W. M. Schmidt, Diophantine Approximation, Mir, 1983 | MR

[20] A Database for Number Fields, (data obrascheniya: 05.05.2018) http://galoisdb.math.upb.de/