Is it true that for any positive integer $m$ there are infinitely many positive integers $n$ such that $\gcd(\lfloor n\sqrt{2}\rfloor, \lfloor n\sqrt{3}\rfloor)=m$?

$\lfloor x \rfloor$ is the floor function of $x$ and $\gcd$ is the greatest common divisor.

In fact we can prove that this result is true for $\gcd(\lfloor n\sqrt{2}\rfloor, n)=m$ (consider a solution $(x_k,y_k)$ to the Pell equation $x^2-2y^2=-1$, such that $x_k>m$ and take $n=my_k$).

Does this result hold if instead of $\sqrt{3}$ we have $\sqrt{k}$, where $k$ is not a perfect square, $k>2$?

Can this result be generalized in any way?

If $a$ and $b$ are positive real numbers such that $\gcd(\lfloor na \rfloor$, $\lfloor nb \rfloor)=m$ holds for infinitely many positive integers $n$ for each $m$, what can be said about $a$ and $b$?