I find that quanta entertains but does not inform. See https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture https://en.wikipedia.org/wiki/Duffin%E

I find that quanta entertains but does not inform. See https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conjecture https://en.wikipedia.org/wiki/Duffin%E2%80%93Schaeffer_conje... for the conjecture in question. Basically it is about under what measures of "a surprisingly good approximation" we would expect to find infinitely many surprisingly good approximations for any random real we pick. Where the definition of a surprisingly good approximation is "within a bound dependent on the denominator". Obviously you can get a better approximation by simply taking a larger denominator. The question is whether you get a good approximation relative to the size of the denominator. We probably aren't surprised to find that with a denominator of size q we can get within 1/q of an arbitrary rational. But we likely are surprised to find that we can get within 1/(100 q^2). In fact there are no approximations that good of the golden ratio (1 + sqrt(5))/2. But it is unusual, almost all real numbers can be approximated that well. But 355/113 is such a good approximation of pi. The conjecture is about the existence, not how to find them. (In practice continued fractions are usually the right way to do it.)