On the linear independence of numbers
The paper contains a proof for the following irrationality criterion. Let \(N_ 0\), \(c_ 1\), \(c_ 2\), \(\tau_ 1\) and \(\tau_ 2>\tau_ 1\) denote positive numbers, and let \(\sigma\) (t) be a monotone-increasing function for all \(t\geq N_ 0\), satisfying \[ \lim_{t\to \infty}\sigma (t)=\infty,\quad \overline{\lim}_{t\to \infty}(\sigma (t+1)/\sigma (t))=1. \] Let \(\theta =(\theta_ 1,...,\theta_ m)\in {\mathbb{R}}^ m\), \(\theta\) \(\neq 0\), and assume that for each natural number \(N>N_ 0\) there exists a linear form \(L_ n(x)=a_{N+1} x_ 1+...+a_{N_ m} x_ m\) with integer coefficients such that \[ \ln \| L_ N\| <\sigma (N),\quad c_ 1 e^{-\tau_ 1 \sigma (N)}\leq | L_ N(\theta)| \leq c_ 2 e^{-\tau_ 2 \sigma (N)} \] (here \(\| L_ N\|\) denotes the length of the vector \((a_{N,1},...,a_{N,m}))\). Then the number of linearly independent (over \({\mathbb{Q}})\) elements of \(\{\theta_ 1,...,\theta_ m\}\) is at least \((\tau_ 1+1)/(1+\tau_ 1-\tau_ 2).\) In particular, if \(\tau_ 2>((m-2)/(m-1))(1+\tau_ 1),\) then the numbers \(\theta_ 1,...,\theta_ m\) are linearly independent over \({\mathbb{Q}}\).
scientific article; zbMATH DE number 3912480