I would like to determine whether or not there exists ${\beta > 0}$ and ${\gamma \geq 2 }$ such that ${ \forall (m_{1},m_{2}) \in \mathbb{Z}^{2} \setminus (0,0) }$, one has the inequality
$$ |m_{1} \!+\! \sqrt{2} \, m_{2}| \geq \frac{\beta}{||(m_{1} , m_{2})||^{\gamma}} \, , $$ where ${|| (m_{1} , m_{2}) ||}$ is a norm on $\mathbb{R}^{2}$.
(I am convinced this should be linked to Diophantine approximation, but I can't get the details to work... And I may be missing the obvious...)
(Context : This comes from KAM Theory, for which in order to conserve regular orbits, the frequencies of motions have to satisfy such diophantine inequalities.)
By Liouville's theorem on Diophantine approximation, (https://en.wikipedia.org/wiki/Diophantine_approximation), there exists a constant $c$ such that $$|\sqrt{2}-{m_1\over m_2}|>{c\over m_2^2}.$$ for every choice of integers $m_1,m_2$ such that $m_2>0$ Consequently, for every choice of integers such that $m_2>0$:
$$|m_2\sqrt{2}-m_1|>{c\over m_2}\geq {c\over\sqrt{m_2^2+m_1^2}}$$
Since this holds for every integer $m_1$, we can replace $m_1$ by $-m_1$ and deduce that: for all pairs $(m_1,m_2)$ such that $m_2>0$ we have:
$$|m_2\sqrt{2}+m_1|> {c\over\sqrt{m_2^2+m_1^2}}$$
Now if $(m_1,m_2)$ is such that $m_2<0$, use the above for the pair $(-m_1,-m_2)$, so that a-posteriori the inequality holds for all pairs of non-zero integers, and the Euclidean norm. As was mentioned above, the result follows now for an arbitrary norm on $\mathbb{R}^2$.