Estimations of the constant of the best simultaneous Diophanite approximations

Yu. A. Basalov

Čebyševskij sbornik, Tome 20 (2019) no. 3, pp. 405-429.

Résumé

This paper is devoted to the development of a new approach for estimating from below the constant of the best Diophantine approximations. The history of this problem dates back to P. G. Dirichlet. Over time, the approaches used to solve this problem have undergone major changes. From algebra (P. G. Dirichlet, A. Hurwitz, F. Furtwengler) this problem has moved into the field geometry of numbers (H. Davenport, J. W. S. Cassels). One cannot fail to note such an interesting component of this problem as the close relationship of diophantine approximations with geometry of numbers in general, and algebraic lattices in particular (J. W. S. Cassels, A. D. Bruno). This provided new opportunities, both for applying the already known results and for application of new approaches to the problem of the best Diophantine approximations (A. D. Bruno, N. G. Moshchevitin). In the mid-twentieth century, H. Davenport found a fundamental relationship between the value of the constant of the best joint Diophantine approximations and critical determinant of a stellar body of a special kind. Later, J. W. S. Cassels went from directly calculating the critical determinant to estimating its value by calculating the largest value of $ V_{n, s} $ – the volume of the parallelepiped centered at the origin with certain properties. This approach allowed us to obtain estimates of the constant of the best joint Diophantine approximations for $ n = 2, 3, 4 $ (see the works of J. W. S. Kassels, T. Cusick, S. Krass). In this paper, based on the approach described above, the estimates $ n = 5 $ and $ n = 6 $ are obtained. The idea of constructing estimates differs from the work of T. Cusick. Using numerical experiments, approximate and then exact values of the estimates $ V_ {n, s} $ were obtained. The proof of these estimates is rather cumbersome and is primarily of technical complexity. Another difference between constant estimates is the ability to generalize them to any dimension. As part of the proof of estimates of the constant of the best Diophantine approximations, we have solved a number of multidimensional optimization problems. In solving them, we used the mathematical package {\ttfamily Wolfram Mathematica} quite actively. These results are an intermediate step for analytical proofs of the estimates of $ V_{n, s} $ and the constant of the best Diophantine approximations $ C_n $ for $ n \geq 3 $. In the process of numerical experiments, interesting information was also obtained on the structure of the values of $ V_{n, s} $. These results are in good agreement 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 little studied and can be of considerable interest both from the point of geometry of numbers and from the point of theory of diophantine approximations.

Détail

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

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