Source: Quanta, Aug 2019
Under what circumstances is it possible to represent irrational numbers that go on forever — like pi — with simple fractions, like 227? The proof establishes that the answer to this very general question turns on the outcome of a single calculation.
Mathematicians had suspected for decades that this simple criterion was the key to understanding when good approximations are available, but they were never able to prove it. Koukoulopoulos and Maynard were able to do so only after they reimagined this problem about numbers in terms of connections between points and lines in a graph — a dramatic shift in perspective.
maybe it’s natural to wonder — if we can’t express irrational numbers exactly, how close can we get? This is the business of rational approximation. Ancient mathematicians, for instance, recognized that the elusive ratio of a circle’s circumference to its diameter can be well approximated by the fraction 227. Later mathematicians discovered an even better and nearly as concise approximation for pi: 355/113.
The key to solving the conjecture has been to find a way to precisely quantify the overlap in the sets of irrational numbers approximated by denominators with many small prime factors in common. For 80 years no one could do it. Koukoulopoulos and Maynard got there by finding a completely different way to look at the problem.
In their new proof, they create a graph out of their denominators — plotting them as points and connecting the points with a line if they share a lot of prime factors. The structure of this graph encodes the overlap between the irrational numbers approximated by each denominator. And while that overlap is hard to assay directly, Koukoulopoulos and Maynard found a way to analyze the structure of the graph using techniques from graph theory — and the information they cared about fell out from there.
“The graph is a visual aid, it’s a very beautiful language in which to think about the problem,” Koukoulopoulos said.
It’s an elegant test that takes a vast question about the nature of rational approximation and boils it down to a single calculable value. By proving that the test holds universally, Koukoulopoulos and Maynard have achieved one of the rarest feats in mathematics: They’ve given a final answer to a foundational concern in their field.
“Their proof is a necessary and sufficient result,” Green said. “I suppose this marks the end of a chapter.”