Positive integers: counterexample to W.M. Schmidt's conjecture

Details Positive integers: counterexample to W.M. Schmidt's conjecture Positive integers: counterexample to W.M. Schmidt's conjecture

2011-08-22

arxiv.org/abs/1108.4435v1

Mathematics

Number Theory

8 pages

Scientific paper

We show that there exist real numbers $\alpha_1,\alpha_2$ linearly
independent over $\mathbb{Z}$ together with 1 such that for every non-zero
integer vector $(m_1,m_2)$ with $m_1\ge 0$ and $m_2\ge 0$ one has
$||m_1\alpha_1+m_2\alpha_2|| \ge 2^{-300} (\max(m_1, m_2))^{-\sigma}$ with
$\sigma = 1.94696^+$.

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