Inhomogeneous inequalities over number fields (Q1328075)

Let \(k\) be a number field and \(S\) a finite collection of places of \(k\) containing all archimedean places. \(k_ S\) denotes the topological product of the completions \(k_ v\), \(v \in S\), and \(\{A_ v\}_{v \in S}\) a collection of \(M \times N\) matrices such that for each \(v \in S\), \(A_ v\) has its entries over \(k_ v\). The \(S\)-system \(\{A_ v\}_{v \in S}\) is called a badly approximable \(S\)-system of linear forms if there is a constant \(\tau>0\) such that \[ \tau < h_ S (\vec x, \vec y)^ N \prod_{v \in S} | A_ v \vec x - \vec y |^ M_ v \] for all \(S\)-integer column vectors \(\vec x \in ({\mathcal O}_ S)^ N\), \(\vec x \neq \vec 0\) and \(\vec y \in ({\mathcal O}_ S)^ M\), where \(h_ S\) is a suitably normalized \(S\)-height. The following two theorems are proved: Theorem 1. Let \(\{A_ v\}_{v \in S}\) be a badly approximable \(S\)- system of dimension \(M \times N\). For each \(v \in S\) suppose that \(\varepsilon_ v \in k_ v\) satisfies \(0<\| \varepsilon_ v \|_ v<1\). Then for any \(\vec \beta = (\beta_ v) \in (k_ S)^ M\), there are vectors \(\vec x \in ({\mathcal O}_ S)^ N\) and \(\vec y \in ({\mathcal O}_ S)^ M\) such that \[ \| A_ v \vec x - \vec y - \vec \beta_ v \|_ v \leq C_ 1 \bigl( k,v, \{A_ v\} \bigr) \| \varepsilon_ v \|^ N_ v \quad \text{and} \quad \| \vec x \|_ v \leq C_ 2 \bigl( k,v,\{A_ v\} \bigr) \| \varepsilon_ v \|_ v^{-M} \] for all places \(v \in S\), where \(\|\;\|_ v\) is a supremum norm, and constants \(C_ 1\) and \(C_ 2\) may be given explicitly. Theorem 2. Let \(A=(A_ v)\) be an \(M \times N\) matrix over the adèle ring \(k_ A\). Suppose that there are idèles \(\varepsilon = (\varepsilon_ v)\) and \(\delta = (\delta_ v)\), with volume \(V(\delta)>1\), so that the system of inequalities \(\| A_ v^ T \vec u - \vec w \|_ v \leq \| \delta_ v \|_ v^{-1}\) for each place \(v\) of \(k\), where \(A^ T_ v\) is the transpose of \(A_ v\), is not solvable with \(\vec u \in k^ M\), \(\vec w\in k^ N\) and \(0< \| \vec u \|_ v \leq \| \varepsilon_ v \|_ v^{-1}\) for each place \(v\) of \(k\). Then for any \(\vec \beta = (\vec \beta_ v) \in (k_ A)^ M\), there are vectors \(\vec x \in k^ N\) and \(\vec y \in k^ M\) so that for each place \(v\) of \(k\), \[ \| A_ v \vec x - \vec y - \vec \beta_ v \|_ v \leq C(k,v,M,N) \| \varepsilon_ v \|_ v \quad \text{and} \quad \| \vec x \|_ v \leq C(k,v,M,N) \| \delta_ v \|_ v, \] where \(C\) is a constant given explicitly. Theorem 1 is a generalization of the classical Khintchine's theorem to number fields. Theorem 2, in a certain sense, may be viewed as a quantitative number field analogue of the classical Kronecker's theorem, and earlier \textit{D. G. Cantor} [Ill. J. Math. 9, 677-700 (1965; Zbl 0142.295)] gave such an analogue, but without an explicit bound on the size of the \(S\)-integer vector \(\vec x\).