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Exponents for three-dimensional simultaneous Diophantine approximations. (English) Zbl 1249.11061

From the author’s summary: Let \(\| \cdot\| \) be the distance from the nearest integer. For \(a=(a_1,a_2,a_3)\in \mathbb R^3\) denote \(\psi_a(t)=\min _{x\in \{1,2,\cdots ,t\}}\max _{i\in \{ 1,2,3\}}\| a_ix\| \). Assume that \(1\), \(a_1\), \(a_2\) and \(a_3\) are linearly independent over \(\mathbb {Z}\). Set \[ \alpha (a)=\sup \{ c>0; \limsup _{t\to \infty }t^c\psi _a(t)<\infty \}, \quad \beta (a)=\sup \{ c>0; \liminf _{t\to \infty }t^c\psi _a(t)<\infty \}. \] Then the author proves that \[ \beta (a)\geq \frac 12\alpha (a)\left(\frac {\alpha (a)}{1-\alpha (a)}+\sqrt {\left(\frac {\alpha (a)}{1-\alpha (a)}\right)^2+\frac {4\alpha (a)}{1-\alpha (a)}}\right). \]

MSC:

11J13 Simultaneous homogeneous approximation, linear forms
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References:

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