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

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

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