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

Abstract

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 $\sigma = 1.94696^+$.