Exponents of Diophantine Approximation in dimension two

Details Exponents of Diophantine Approximation in dimension two Exponents of Diophantine Approximation in dimension two

2006-11-12

arxiv.org/abs/math/0611352v1

Mathematics

Number Theory

Scientific paper

Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents which measure the quality of approximation to $\Theta$ both by rational points and by rational lines. The two ``uniform'' components of $\Omega(\Theta)$ are related by an equation, due to Jarn{\'\i}k, and the four exponents satisfy two inequalities which refine Khintchine's transference principle. Conversely, we show that for any quadruple $\Omega$ fulfilling these necessary conditions, there exists a point $\Theta\in \bR^2$ for which $\Omega(\Theta) =\Omega$.

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