Simultaneous approximation lattices of p-adic numbers (Q1262888)

Let \(\alpha_ 1,...,\alpha_ n\in {\mathbb{Q}}_ p\) (with p a prime number). The authors call a p-adic simultaneous approximation of \((\alpha_ 1,...,\alpha_ n)\) of order m an \((n+1)\)-tuple \((Q,Q_ 1,...,Q_ n)\in {\mathbb{Z}}^{n+1}\) such that \[ | Q-Q_ 1\alpha_ 1- ...-Q_ n\alpha_ n| \leq p^{-m} \] and they denote by \(\Gamma_ m\) the set of these \((n+1)\)-tuples. The sequence \((\Gamma_ m)\) is called the sequence of simultaneous approximation and is clearly a decreasing sequence. This sequence is said to be periodic if there exists \(k\in {\mathbb{N}}\) and a linear mapping from \({\mathbb{R}}^{n+1}\) to \({\mathbb{R}}^{n+1}\) such that \(f(\Gamma_ m)=\Gamma_{m+k}\) (at least for \(m\geq m_ 0).\) They show that if \(\alpha_ 1,...,\alpha_ n\in {\mathbb{Z}}_ p\cap {\mathbb{Q}}\) then the sequence of simultaneous approximation is periodic.If \(\alpha_ 1,...,\alpha_ n\in {\mathbb{Z}}_ p\) and if \(1,\alpha_ 1,...,\alpha_ n\) are \({\mathbb{Q}}\)-linearly independent there are infinitely many \((n+1)\)-tuples \((Q,Q_ 1,...,Q_ n)\) in \({\mathbb{Z}}^{n+1}\) such that \[ | Q-Q_ 1\alpha_ 1-...-Q_ n\alpha_ n| \leq \frac{2(n+2)}{\max (Q,Q_ 1,...,Q_ n)^{n+1}}. \]