Abstract
This paper presents the main stages in the study of the distribution of fractional parts of polynomial values whose arguments lie in short intervals. The research is based on methods of analytic number theory, particularly on estimates of trigonometric sums, starting from classical results by Gauss and Weyl and extending to modern developments. It is shown that the key tool for studying the distribution of fractional parts of the values of various polynomials, whose arguments take values from short intervals, is the method of trigonometric sums, which makes it possible to obtain non-trivial estimates and to establish the uniform distribution of sequences modulo one. The paper reviews estimates of rational and irrational trigonometric sums, including contributions by Mordell, Weyl, Hua Loo-Keng, and Vinogradov. Special attention is paid to short Weyl sums and their applications to the study of polynomial values over short intervals. The results of Tajik mathematicians, especially Z.Kh. Rakhmonov, N.B. Ozodbekova, S.N. Ismatov and F.Z. Rakhmonov are discussed, focusing on bounds for short sums and their applications to uniform distribution problems. The obtained results demonstrate substantial progress in analytic number theory and its applications to distribution problems. The study confirms the central role of trigonometric sums as a powerful and versatile tool in modern number theory.