The deep recesses of the number line are not as forbidding as they might seem. That’s one consequence of a major new proof about how complicated numbers yield to simple approximations.
The proof resolves a nearly 80-year-old problem known as the Duffin-Schaeffer conjecture. In doing so, it provides a final answer to a question that has preoccupied mathematicians since ancient times: 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.
“There’s a simple criterion for whether you can approximate virtually every number or virtually no numbers,” said James Maynard of the University of Oxford, co-author of the proof with Dimitris Koukoulopoulos of the University of Montreal.
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.
“They had what I’d say was a great deal of self-confidence, which was obviously justified, to go down the path they went down,” said Jeffrey Vaaler of the University of Texas, Austin, who contributed important earlier results on the Duffin-Schaeffer conjecture. “It’s a beautiful piece of work.”
From quanta magazine, and I should go through this. Hopefully I will.
This proof is closely related to the fractional approximation algorithm of Huritz, this:
Observations like these have led mathematicians to set up a hierarchy among irrational numbers, according to how difficult they are to approximate with rationals. It is in this sense that one irrational is more irrational than another. To make the criterion precise, we start from the following fact:
Hurwitz' Theorem: Every number has infinitely many rational approximations p/q, where the approximation p/q has error less than 1/q2.
This is a sandbox issue because we approximate the aggregate S/L ratio with whole number fractions. This proof generalizes Horitz, I think.
Also of note, it uses graph theory for proofs. This also shows up in 'Good Will Hunting' in which Matt Damon take a graph on the chalk board, writes the equivalent connecting equations, and completes a proof. I guessed that advances in number theory would come from graph theory. There approach was sandbox, I think. They create the graph of denominators connected with their co-prime partners, then show that the garph compresses uniformly. (I think, I just skimmed over a few introductary paragraphs so far, take this with a grain).
This proof also includes round off error, how does specifying round off error change the density of solutions. This is a difficult read, one in which a art timer like me as to constantly refer back to basic Wiki texts.
The other thing is that Quant is on the ball here, following some historic evens closely in math and physics, in this case the Quant author was Kevin Hartnett. Great magazine for the not quite so clueless folks who follow this stuff.
No comments:
Post a Comment