On a decomposition of integer vectors. I (Q2739299)

It is proved that for every non-zero vector \(\mathbf n=(n_1,n_2,n_3)\in\mathbb Z^3\) with height \(h(\mathbf n)=\max|n_i|\), there exist linearly independent vectors \(\mathbf p,\mathbf q\in\mathbb Z^3\) such that \(\mathbf n=u\mathbf p+v\mathbf q\), where \(u,v\in\mathbb Z\) and \(h(\mathbf p)h(\mathbf q)\leq C(\mathbf n)(h(\mathbf n))^{\frac 12}\). Here \(C(\mathbf n)\) is an explicitly given function satisfying \(1<C(\mathbf n)<2/\sqrt 3\). The proof is based on ideas of \textit{A. Schinzel} [Bull. Pol. Acad. Sci., Math. 35, 155-159 (1987; Zbl 0639.10023)]. It gives a key for solving similar questions in higher dimensions.