BRIEF MESSAGE

Note on a theorem of Davenport

Ke Gong

Henan University (Kaifeng, P. R. China)

Abstract: Let $\Lambda$ be a $n$-dimensional lattice, and $c_1,\ldots,c_{n-1}$ be any $n-1$ vectors in $n$-dimensional real Euclidean space. We show that there exists a basis $\alpha_1,\ldots,\alpha_n$ of $\mathsf\Lambda$ such that
$$ |\alpha_i-Nc_i|=O(\log^2N),\leqslant (1\leqslant i\leqslant n-1) $$
holds for any real number $N\ge 2$, where the constant implied by the $O$ symbol depends only on $\Lambda$ and $c_1,\ldots,c_{n-1}$.

Keywords: Lattice, basis, approximation, combinatorial sieve.

UDC: 511

Language: English

DOI: 10.22405/2226-8383-2018-22-2-484-489