I am trying the find a pair $(a,b)\in \mathbb R^2$ such that $\inf_{m,n\in \mathbb Z}\left|(m+\sqrt{3}n+a)(\sqrt{2}m+n+b) \right|>0$
For $a=b=\frac{1}{2}$, I speculate that the infimum is nonzero. By Dirichlet's theorem, we have $|\sqrt{2} m+n+\frac{1}{2}|\lesssim |\frac{1}{n}|$ for infinitely many $m,n$ and $\sqrt{2} m \asymp -n$. So $|m+\sqrt{3}n+a|\sim |n|$. But I don't see why $$\inf_{m,n\in \mathbb Z}\left|(m+\sqrt{3}n+\frac{1}{2})(\sqrt{2}m+n+\frac{1}{2}) \right|>0$$
My question is about finding such a pair explicitly, but if you have a non-constructive proof showing that such pairs exist, please also let me know.