What is $$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N} |2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$ where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to know how small a real $p$ can be so that for some real $c>0$ and all natural $n,k$ we have $$|2\sqrt{3n}-9\pi/4-k\pi|\ge \frac c{n^p}.$$ One may note that $p_*\ge1/2$, since $\sqrt{n+1}-\sqrt n<n^{-1/2}$ and hence $\inf_{k\in\mathbb N} |2\sqrt{3n}-9\pi/4-k\pi|<\sqrt3\,n^{-1/2}$ for infinitely many $n$, so that $\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N} |2\sqrt{3n}-9\pi/4-k\pi|>0\big\}=0$ if $p<1/2$.
This question is related to this previous one.
As fedja noted in a comment, this problem seems similar to the problem of determining the irrationality measure of $\pi$, and thus may be just too hard.
Actually, for my purposes it would be enough to know how $p_*$ compares with $9/4$: Is $p_*$ greater than, less than, or equal to $9/4$?
