How to find this Littlewood conjecture estimate?

Question

Suppose Littlewood's conjecture is true. So we have $$\liminf_{n\rightarrow\infty}n\|n\alpha\|\|n\beta\|=0$$ for all $\alpha$ and $\beta$. In particular, for some $n$ we have $$n\|n\alpha\|\|n\beta\| < 10^{-30}$$ If finding explicitly such an $n$ is not feasible, can an estimate of it's size be found? I am most interested in the case when $\alpha=\sqrt 2$ and $\beta=\sqrt 3$ if that helps.