On the history of estimates of the constant of the best joint diophanite approximations

Yu. A. Basalov

Čebyševskij sbornik, Tome 19 (2018) no. 2, pp. 389-406.

Résumé

This paper presents a historical review of the results on the problem of estimating a constant best joint 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 itself rich history, ascending to P. G. Dirichlet. We will focus more on the approach of H. Davenport, based on the connection of Diophantine approximations with the geometry of numbers.The first part provides an overview of results obtained for $ n = 1 $ and $ n = 2 $ real numbers. Historically, estimates for $ n = 1 $ are based on the theory of continued fractions and the most significant is A. Hurwitz's estimate obtained in 1891. For $ n = 2 $ the basis of known estimates is mathematical apparatus of linear algebra (F. Furtwangler), geometry of numbers (H. Davenport, J. W. S. Cassels) and the results of multidimensional generalizations of continued fractions (V. Adams, T. Cusick).The second part is devoted to one of the first general estimates from below obtained in 1929 by F. Furtwangler. He constructed estimates of the discriminants of algebraic fields, which lead to an estimate of the quality of the approximation of $ n $ real numbers by rational, which in turn results in estimating the constant best joint diophantine approximations.The third part outlines the most fundamental of the currently known assessments obtained H. Davenport and then refined by J. W. S. Cassels. H. Davenport discovered the connection between the value critical determinant of lattices and the estimating of some forms. In the particular case it allows to get the value of the constant best joint diophantine approximations by calculating the critical determinant of the special lattice. However the calculation of critical determinants for lattices of this type is a difficult task. Therefore, J. W. S. Cassells switched from the direct calculation of the critical determinant to the estimating of its value. This approach proved to be quite fruitful and allowing us to obtain estimates of the constant best joint diophantine approximations for $ n = 2, 3, 4 $.The fourth part gives an overview of the well-known estimates from below for $ n > 2 $. These results are based on the use of the aforementioned approach of J. W. S. Cassels. It is worth noting that assessments of this kind are quite a complicated computational problem and in each case the solution of this problem required the use of new approaches.In the last part we present a review of some well-known estimates of the constant best diophantine approximations from above. Although this problem is not the main topic of this articles, but considerable interest is the comparison of approaches in assessing constants of the best joint diophantine approximations from above and below. The first estimate from above was obtained by H. Minkowski in 1896 using the geometry of numbers. H. F. Blichfeldt introducing the concept of the fundamental parallelepiped in 1914 improved the result of H. Minkowski. Later the approach of H. F. Blichfeldt received development in the works of P. Mullender, W. G. Spohn, W. G. Nowak.

Détail

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

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