About the method of estimating critical determinants within the question of the estimation of the constant of simultaneous diophantine approximations

Yu. A. Basalov

Čebyševskij sbornik, Tome 20 (2019) no. 2, pp. 22-38

Résumé

This paper is devoted to the estimation of the constant of sumultaneous Diophantine approximations for $ n $ real numbers. The approach developed by H. Davenport and J. W. S. Cassels. H. Davenport discovered the connection between the value of the critical determinant of a star body and the estimation of some forms. In the particular case, this allows calculating the critical determinant of the $ (n + 1) $-dimensional star body of Davenport $$ \mathbb {F}_{n}: | x_0 | \max \limits_{1 \leq i \leq n} | x_i |^n 1, $$ get the value of the constant of joint Diophantine approximations. However, the calculation of critical determinants for bodies of this type is a difficult task. Therefore, J. W. S. Cassels moved from directly calculating the critical determinant, to estimating its value. For this, he used the estimate of the largest value of $ V_{n, s} $ – the volume of a parallelepiped centered at the origin of coordinates located inside the $ (n + 1) $-dimensional star body $$ \mathbb {F}_{n, s}: f_{n, s} = \frac 1 {2^s} \prod \limits_{i = 1}^{s} | x_i^2 + x_{s + i}^2 | \prod \limits_{i = 2s + 1}^{n} | x_i | 1. $$ These results reduce the problem of estimating the constant of joint Diophantine approximations to an estimate of the volume of the largest parallelepiped $ V_{n, s} $. Earlier, estimates for $ V_{n, s} $ were obtained in the works of J. W. S. Cassels, T. Cusick, S. Krass. This paper is devoted to methods of forming hypotheses about the values of $ V_{n, s} $ based on the results of numerical experiments. The article outlines the approach to obtaining parallelepipeds contained within a star body and possessing the largest volume. This approach combines the use of both numerical and analytical methods.

Détail

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