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On the linear independence of numbers. (English. Russian original) Zbl 0572.10027
Mosc. Univ. Math. Bull. 40, No. 1, 69-74 (1985); translation from Vestn. Mosk. Univ., Ser. I 1985, No. 1, 46-49 (1985).
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 $${\mathbb{Q}}$$.
Reviewer: K.Väänänen

##### MSC:
 11J81 Transcendence (general theory)
##### Keywords:
linear independence; irrationality criterion