
New Proof Solves 80-Year-Old Irrational Number Problem - chris_overseas
https://www.scientificamerican.com/article/new-proof-solves-80-year-old-irrational-number-problem/
======
xelxebar
For those that just want the meat, this is about the Duffin-Schaeffer
conjecture [0]. Here's the relevant paper:

[https://arxiv.org/abs/1907.04593](https://arxiv.org/abs/1907.04593)

For those that want a little more, the Duffin-Schaeffer conjecture is about
rational approximations of real numbers. The setup is this:

Given some real number, we're concerned with its rational approximations. To
this we choose some function that assigns an "acceptable error" for every
possible denominator.

Thus, we want to look at the set of "acceptable approximations." A priori, we
can intuit that if our errors are too strict, we might not have any---or only
a handful---that meet our criteria. The Duffin-Schaeffer makes this intuition
precise, giving a condition that tells us whether our set of acceptable
approximations will be finite or infinite.

[0]:[https://en.wikipedia.org/wiki/Duffin-
Schaeffer_conjecture](https://en.wikipedia.org/wiki/Duffin-
Schaeffer_conjecture)

------
signalsmith
Slightly off-topic, but for anyone who hasn't encountered them: using
continued fractions
([https://en.wikipedia.org/wiki/Continued_fraction#Best_ration...](https://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations)),
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.

~~~
contravariant
Note that that sequence of approximations of pi is not the _entire_ sequence
of optimal approximations. That one requires a bit more work.

It is _a_ sequence of optimal approximations though.

