On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post.

I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / number-theoretic equation $f(a,b)=0$.

Let $S$ be the set $$S = \{(a,b)|f(a,b)=0 \}.$$

Suppose that I have bounds for the sum $$L(a,b) \leq \frac{a}{b} + \frac{b}{a} \leq U(a,b).$$

If I want to prove that $S$ is empty, one way is to prove that the ratio $a/b$ (or equivalently, $b/a$) is actually irrational.

Since we know that, in general $$\frac{a}{b} + \frac{b}{a} \rightarrow \infty,$$ does the upper bound $U(a,b)$ guarantee an irrationality proof?

If anybody can point out to me a proof (in Diophantine approximation) that proceeds along similar lines, I will most certainly appreciate it.

• You ask whether $\lim (a/b + b/a) = \infty$ ensures irrationality. I think the answer is no. Only $\liminf (a/b + b/a) = \infty$ ensures irrationality. – user98186 Dec 26 '15 at 21:11
• @NimaBavari, can you give specific examples for both of your claims that $\lim(a/b + b/a) = \infty \not{\Rightarrow}$ irrationality, and $\lim \inf(a/b + b/a) = \infty \implies$ irrationality, in an actual answer, so that I may be able to review and accept it, if needs be? Thanks! – Jose Arnaldo Bebita-Dris Dec 27 '15 at 10:39
• Welcome! I don't know if I can rigorously prove my claim but it's very intuitive. If I recall right, it is a standard procedure when working with Shnirelmann density: not only one element of a set but all of them should fall between a specific interval so that a certain property be satisfied. I can comment like this but I'm not sure if I can formally and elaborately convey my idea. – user98186 Dec 27 '15 at 18:00