From Galois theory, we know that $[\mathbb{Q}(\sqrt{2}, \sqrt{3}, \dots \sqrt{p_k}) : \mathbb{Q}] = 2^k$. Suppose I plug in rational approximations to the square roots, then of course the classical theorem would no longer hold since rationals are linearly dependent over rationals. Is there a way to define some sort of "bounded linear dependence" -- maybe by enforcing magnitude conditions on linear dependence and declaring a set of rational numbers to be linearly independent if no such dependence exists (at this point I'm not even sure if this is well-defined) -- What is the right notion to capture this requirement and has this been investigated before? My gut feeling is that this should have something to do with Diophantine approximation, but since I'm unfamiliar with the area, I don't know where to start looking.

I admit my question is a little vague, but I'd be grateful for any relevant pointers!