About the Russian scientific school of diophantine approximations

Yu. A. Basalov

Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 388-403.

Résumé

The theory of diophantine approximations, as a branch of mathematics, began to take shape in the 19th century. A significant contribution to its development was made by Russian and Soviet mathematicians. In this paper, we give a historical review of some results in the field of Diophantine approximations obtained by the Russian scientific school of number theory. One of the first, P. L. Chebyshev became interested in the problems of the theory of diophantine approximations in the second half of the 19th century. These studies were continued by his students A. N. Korkin and E. N. Zolotarev. In 1880 academician A. A. Markov (the student of A. N. Korkin) in his master thesis brilliantly solved the problem of describing classes of poorly approximating indefinite quadratic forms. Another student of P. L. Chebyshev — G. F. Voronoj, along with G. Minkowski, laid the foundations for a new, closely related to Diophantine approximations section of mathematics — the geometry of numbers. A. I. Hinchin made a significant contribution to the development of the metric theory of continued fractions. In 1936, he obtained the Khinchin constant — the value of the geometric mean of the decomposition into a continued fraction, for almost all real numbers. The awesomeness of this fact is noted by mathematicians around the world. A significant contribution to the development of the metric theory of diophantine approximations belongs to Belarusian mathematicians. In 1964 V. G. Sprindzhuk obtained a proof of the hypothesis on the measure of the set of $S$-numbers. Research in this area was continued by V. I. Bernik. Interesting results in the field of geometry of numbers and the approximation properties of algebraic numbers were obtained in the 70–80s of the XX century by B. F. Skubenko. In particular his work presents an estimate of the constant of the best Diophantine approximations for the two-dimensional case. Research in the field of approximation of real numbers and the theory of continued fractions was continued in the 1990–2010s by N. G. Moshchevitin, O. N. German, A. D. Bruno, N. M. Dobrovol'skii and N. N. Dobrovol'skii.

Détail

Keywords: history of diophantine approximations, Russian school of number theory.

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