A new version of the ergodic method of Yu. V. Linnik in number theory

A new version of a known method of analytic number theory is developed. This version is demonstrated with a very simple model example — the problem on the asymptotically uniform distribution of integer points of a sphere with respect to a given modulus.This paper is a reworking of §5 of survey 6].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 50, pp. 179–186, 1975.