Liouville’s approximation theorem says:
If $\alpha$ is algebraic irrational of degree $n\geq2$, then there exists a real number $A$ such that for all integers $p,q$ with $q>0$ and $$\left|\alpha-\frac{p}{q}\right|>\frac{A}{q^n}.$$
I am interested in the constant $A$, is there a best $A=A(n,\alpha)$ in the follows sense $$A(n,\alpha)=:\inf_{p\geq0,q\geq1}q^n\cdot\left|\alpha-\frac{p}{q}\right| =\inf_{p\geq0,q\geq1}q^{n-1}\cdot|q\alpha-p|.$$ There are some partial result in the special cases: for $n=2$, $$\left|\sqrt{2}-\frac{p}{q}\right|>\frac{1}{3q^2},\quad \left|\sqrt{3}-\frac{p}{q}\right|>\frac{1}{5q^2}，$$ and (with $\alpha=\sqrt d$) $$\left|\sqrt{d}-\dfrac{p}{q}\right|\geq\frac{1}{q^2(2\sqrt{d}+1)},\quad \left|\sqrt{d}-\dfrac{p}{q}\right|\geq\dfrac{\sqrt{d+1}-\sqrt{d}}{q^2}.$$
In the case $\alpha=\sqrt2,n=2$, it seems $$\inf_{p\geq0,q\geq1}q\left|\sqrt{2}q-p\right| =\frac{2}{3+2\sqrt{2}}\ \text{when}\ p=3,q=2.$$
In the case $\alpha=\sqrt3,n=2$, it seems $$\inf_{p\geq0,q\geq1}q\left|\sqrt{3}q-p\right| =\frac{1}{\sqrt3+2}\ \text{when}\ p=2,q=1.$$
If there are some colsed form for the best constant $A(n,\alpha)$ or in some special cases such as $n=2$. Any help and hints will welcome, thanks!