Let $\|x\|$ denotes the distance between $x\in\mathbb{R}$ and the nearest integer to $x$.
It's an easy application of Dirichlet pigeonhole principle to prove that for any $N\geq 2$ there exist integers $m,n$ such that $|m|,|n|\leq N$, $(m,n)\ne(0,0)$ and $$\|m\sqrt2+n\sqrt3\|<\frac{1}{N^2}.$$
I'm interested in the lower bound of $\|m\sqrt2+n\sqrt3\|$. For example, is it true that for some (or any?) $\varepsilon>0$ and $c=c(\varepsilon)>0$ we have $$\|m\sqrt2+n\sqrt3\|>\frac{c}{N^{2+\varepsilon}}$$ for all $|m|,|n|\leq N$, $(m,n)\ne(0,0)$?
I have a feeling it's something known (or have been studied) in number theory. If so, a nice reference would be appreciated.
By Roth's Theorem, since $\sqrt{\frac{2}{3}}$ is algebraic, there are only finitely many $\frac{m}{n} \in \mathbb{Q}$ such that $$\left|\sqrt{\frac{2}{3}}- \frac{m}{n} \right| < \frac{1}{n^{2+ \epsilon}}.$$ Thus, there is a $c= c(\epsilon)$ such that $$\left|\sqrt{\frac{2}{3}}- \frac{m}{n} \right| > \frac{c}{n^{2+ \epsilon}}$$ for all $m$ and $n$. This implies your statement.