chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2021-22-2-484-489cheb-1014Research ArticleКраткие сообщенияЗамечание к теореме ДавенпортаNote on a theorem of DavenportГонгКеGongKe

доктор наук, доцент

doctor of science, associate professor

kg@henu.edu.cnХэнаньский университетКитайHenan UniversityChina202102062021222484489Copyright © Гонг К., 20212021Гонг К.Gong K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1014

Пусть Λ-𝑛-мерная решетка, а 𝑐1, . . . , 𝑐𝑛−1 - любые 𝑛 − 1 векторов в 𝑛-мерном вещественном евклидовом пространстве. В работе доказано существование базиса 𝛼1, . . . ,𝛼𝑛 решётки Λ такого, что неравенство

$$|𝛼𝑖 − 𝑁𝑐𝑖| = 𝑂(log^2 𝑁), (1 <= 𝑖 <= 𝑛 − 1)$$

имеет место для любого вещественного 𝑁 > 2, где константа в знаке 𝑂 зависит лишь от Λ и 𝑐1, . . . , 𝑐𝑛−1.

Let Λ be a 𝑛-dimensional lattice, and 𝑐1, . . . , 𝑐𝑛−1 be any 𝑛 − 1 vectors in 𝑛-dimensional real Euclidean space. We show that there exists a basis 𝛼1, . . . ,𝛼𝑛 of Λ such that

$$|𝛼𝑖 − 𝑁𝑐𝑖| = 𝑂(log^2 𝑁), (1 <= 𝑖 <= 𝑛 − 1)$$

holds for any real number 𝑁 > 2, where the constant implied by the 𝑂 symbol depends only on Λ and 𝑐1, . . . , 𝑐𝑛−1.

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The authors declare that there are no conflicts of interest present.