Here's a demonstration I made recently, of best rational approximations: https://shreevatsa.net/fraction/best https://shreevatsa.net/fraction/best — much of thi

Here's a demonstration I made recently, of best rational approximations: https://shreevatsa.net/fraction/best https://shreevatsa.net/fraction/best — much of this Quanta magazine article sets up background on such approximations, so playing with the numbers on this web page may help build some intuition about the problem. In particular, the Duffin-Schaeffer conjecture that has just been proved involves asking about the set of irrational numbers that are approximated to within an error of f(q) for each denominator q, while the webpage above simply finds the best possible approximations for a single fixed irrational number (well, rational…) from the (semi-)convergents of its continued fraction — so any denominator is allowed and the error (for convergents) turns out to be the Dirichlet bound 1/q^2. (For best results, choose a really large bound on the denominator and only ask for the “bestest” rational approximations. The source code is just a single HTML file and a bunch of plain JavaScript files to do the computation, so if you can improve it, please do so and let me know: https://gitlab.com/svat/web-fraction/tree/master/website https://gitlab.com/svat/web-fraction/tree/master/website .)