Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
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 22/7? 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.”
The Ether of Arithmetic
Rational numbers are the easy numbers. They include the counting numbers and all other numbers that can be written as fractions.
This amenability to being written down makes rational numbers the ones we know best. But rational numbers are actually rare among all numbers. The vast majority are irrational numbers, never-ending decimals that cannot be written as fractions. A select few are important enough to have earned symbolic representations, such as pi, e and the square root of 2. The rest can’t even be named. They are everywhere but untouchable, the ether of arithmetic.
So 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 22/7. Later mathematicians discovered an even better and nearly as concise approximation for pi: 355/113.
“It’s hard to write down what pi is,” said Ben Green of Oxford. “What people have tried to do is to find explicit approximations to pi, and one common way of doing that is with rationals.”
In 1837 the mathematician Gustav Lejeune Dirichlet found a rule for how well irrational numbers can be approximated by rational ones. It’s easy to find approximations so long as you’re not too particular about the error. But Dirichlet proved a straightforward relationship between fractions, irrational numbers and the errors separating the two.
He proved that for every irrational number, there exist infinitely many fractions that approximate the number evermore closely. Specifically, the error of each fraction is no more than 1 divided by the square of the denominator. So the fraction 22/7, for example, approximates pi to within 1/72, or 1/49. The fraction 355/113 gets within 1/1132, or 1/12,769. Dirichlet proved that there is an infinite number of fractions that draw closer and closer to pi as the denominator of the fraction increases.
“It’s a rather beautiful and remarkable thing that you can always approximate a real number by a fraction and the error is no more than 1 over [the denominator squared],” said Andrew Granville of the University of Montreal.
Dirichlet’s discovery was, in a sense, a narrow statement about rational approximation. It said that you can find infinitely many approximating fractions for each irrational number if your denominators can be any whole number, and if you’re willing to accept an error that’s 1 over the denominator squared. But what if you want your denominators to be drawn from some (still infinite) subset of the whole numbers, like all prime numbers, or all perfect squares? And what if you want your approximation error to be 0.00001, or any other values you might choose? Will you succeed at producing infinitely many approximating fractions under such specific conditions?
The Duffin-Schaeffer conjecture is an attempt to provide the most general possible framework for thinking about rational approximation. In 1941 the mathematicians R.J. Duffin and A.C. Schaeffer imagined the following scenario. First, choose an infinitely long list of denominators. This could be anything you want: all odd numbers, all numbers that are multiples of 10, or the infinite list of prime numbers.
Second, for each of the numbers in your list, choose how closely you’d like to approximate an irrational number. Intuition tells you that if you give yourself very generous error allowances, you’re more likely to be able to pull off the approximation. If you give yourself less leeway, it will be harder. “Any sequence can work provided you leave enough room,” Koukoulopoulos said.
Now, given the parameters you’ve set up — the numbers in your sequence and the defined error terms — you want to know: Can I find infinitely many fractions that approximate all irrational numbers?
The conjecture provides a mathematical function to evaluate this question. Your parameters go in as inputs. Its outcome could go one of two ways. Duffin and Schaeffer conjectured that those two outcomes correspond exactly to whether your sequence can approximate virtually all irrational numbers with the demanded precision, or virtually none. (It’s “virtually” all or none because for any set of denominators, there will always be a negligible number of outlier irrational numbers that can or can’t be well approximated.)
“You get virtually everything or you get virtually nothing. There’s no middle ground at all,” Maynard said.
It was an extremely general statement that tried to characterize the warp and weft of rational approximation. The criterion that Duffin and Schaeffer proposed felt correct to mathematicians. Yet proving that the binary outcome of this function is all you need to know whether your approximations work — that was much harder.
Proving the Duffin-Schaeffer conjecture is really about understanding exactly how much mileage you’re getting out of each of your available denominators. To see this, it’s useful to think about a scaled-down version of the problem.
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Imagine that you want to approximate all irrational numbers between 0 and 1. And imagine that your available denominators are the counting numbers 1 to 10. The list of possible fractions is pretty long: First 1/1, then 1/2 and 2/2, then 1/3, 2/3, 3/3 and so on up to 9/10 and 10/10. Yet not all of these fractions are useful.
The fraction 2/10 is the same as 1/5, for example, and 5/10 covers the same ground as 1/2, 2/4, 3/6 and 4/8. Prior to the Duffin-Schaeffer conjecture, a mathematician named Aleksandr Khinchin had formulated a similarly sweeping statement about rational approximation. But his theorem didn’t account for the fact that equivalent fractions should only count once.