Abstract:
This article is dedicated to the 110th anniversary of the birth of the outstanding world — famous mathematician Academician Yu. V. Linnik and his discrete ergodic method. First, biographical information about Yu. V. Linnik is given.
Then, after a brief presentation of the necessary information from the arithmetic of quaternions including the theory of quaternion rotations constructed by B. A. Venkov, the very idea of the discrete ergodic method (hereinafter DEM), belonging to Yu. V. Linnik, is considered.
The next part of the article is devoted to the presentation of the ergodic theorem in the case of quaternions and its application to the question of the asymptotics of integer points over regions on a sphere with increasing radius.
After this applications of the DEM to indefinite ternary quadratic forms corresponding to cases of integer point distributions on hyperboloids, using second-order matrix arithmetic instead of quaternions.
The article concludes with a statement of some unsolved problems related of DEM and a list of references.
Keywords:Yuri Vladimirovich Linnik, discrete ergodic method (DEM), quaternion matrix of the second order, ergodic theorem, asymptotic distribution of integer points of spheres and hyperboloids.
Fulltext:
PDF file (619 kB)