Note on a theorem of Davenport
https://doi.org/10.22405/2226-8383-2021-22-2-484-489
Abstract
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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Review
For citations:
Gong K. Note on a theorem of Davenport. Chebyshevskii Sbornik. 2021;22(2):484-489. (In Russ.) https://doi.org/10.22405/2226-8383-2021-22-2-484-489
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