Take $N=M=2$ and $\mathbb{x} :=(1:x)$, $\mathbb{y} := (1:m/n)$ in Thm. 3, with the projective distance defined in (1.2). Your question follows since $|\mathbb{x} \wedge \mathbb{y}| = |x-m/n|$. So you may quote the paper of Choi and Vaaler as a reference to a much more general setting.

Better: see the original source, K. K. Choi and J. D. Vaaler, Diophantine approximation in projective space: cecm.sfu.ca/~choi/paper/metric.pdf .

See section 2.8 in Heights in diophantine geometry by Bombieri and Gubler, particularly Prop. 2.8.19.

Actually, see here: mathoverflow.net/questions/38971/…

You are welcome! And actually, upon looking at a paper of Stewart from 1980, I realize that I was wrong about effectivity: an explicit lower bound is actually possible with Baker's method. I will edit.

Yes, this is the difference between inseparable and separable rational connectivity (which only arises in positive characteristic, of course). The proposition remains true upon adding the qualifier "separably" in front of "uniruled" and "rationally connected"; this is Theorems IV 1.9 and IV 3.7 in Kollar's book. The example you give is not separably uniruled.

But why do you write "in characteristic zero?" Over any uncountable field, uniruledness simply means that there is a rational curve through any general point; rational connectivity is in turn the stronger property that there is a rational curve through any two general points.

I was just about to remark the same thing :-). But please do not remove your answer as it contains valuable references to the literature! (My answer gives, in turn, the complementary reference to Sandling's paper.)

Plainly, to paraphrase a line from an article (Geometry and physics, Phil. Trans. R. Soc. A 2010, highly recommended) by Atiyah, Dijkgraaf, and Hitchin, dualities in physics, including the one that underlies the enumeration formula you mention, are often [always?] "captured by a generating function that allows two different expansions." This is true of other mathematical formulas as well. In this way generating series are more than just a formal book keeping device for recurrence relations among coefficients. In any case they fully deserve to be called functions rather than power series.