## 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

### Keywords:

Diophantine approximations; Diophantine exponent
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### References:

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