Slightly off-topic, but for anyone who hasn't encountered them: using continued fractions (https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...), you can easily get the sequence of optimal approximations - where "optimal" means "better than anything else with a smaller denominator".

(e.g. pi -> 3, 22/7, 333/106, 355/113, ...)

This paper is about a different situation (Duffin-Schaeffer), where you don't just want the minimum denominator, but instead have more structured/custom constraints for whether a approximation passes/fails, based on the denominator.