Certain properties of ``good'' diophantine approximations
The author studies certain approximation properties of lattice points. Let \(\Lambda\subseteq\mathbb{R}^n\) be a complete lattice with determinant \(V_n(\Lambda)= D\) and vectors \({\mathbf v}_i= (x_{i1},\dots, x_{in})\in \Lambda\), \(i=1,\dots, n\). Suppose that the numbers \(a_0,Y,X_1,\dots, X_n\), \(A_1,\dots, A_n>0\) and \(\alpha= (\alpha_1,\dots,\alpha_n)\in \mathbb{C}^n\) are given. The author proves that if the inequalities \[ \max_{1\leq j\leq n}|\alpha_j|A_j\geq a_0,\quad \max_{1\leq j\leq n}|x_{ij}|A_j\leq X_i,\quad i=1,\dots,n;\tag{a} \] or \[ \Biggl(\sum^n_{j=1} |\alpha_j|^2 A^2_j\Biggr)^{1/2}\geq a_0,\quad \Biggl(\sum^n_{j= 1}|x_{ij}|^2 A^2_j\Biggr)^{1/2}\leq X_i,\quad i=1,\dots, n,\tag{b} \] hold, and the inequalities \[ Y>(X_1 A_1)\cdots (X_n A_n),\quad \Biggl|\sum^n_{j= 1}\alpha_j x_{ij}\Biggr|\leq a_0cDX_i Y^{-1},\quad i=1,\dots, n \] are satisfied, where \(c= n^{-n/2}\) if (a) holds, and \(c= n^{-3/2}\) if (b) holds, then \({\mathbf v}_1,\dots,{\mathbf v}_n\) are linearly dependent over \(\mathbb{Z}\). He also gives the coordinate-free variation of this result and its generalization to algebraic number fields. As an illustration of the application of these results, the author establishes a refined lower estimate for the Hermite interpolation polynomial, which is often useful in transcendental number theory.
approximation properties of lattice points
scientific article; zbMATH DE number 1028452