Fractional Approximations for Irrational Numbers

Pi is an irrational number, which is to say that it is a real number than cannot be precisely written as the ratio of two integers in a simple fraction.

That’s not to say that you cannot reasonably approximate the value of pi with such a fraction however – it’s just a question of how much error you’re willing to live with by doing so. If the results of your math would be okay if you only approximated pi to just two decimal places, which in decimal form is 3.14, you could substitute the rational fraction 22/7. If you needed up to six decimal places of rounded-up precision, 3.141593, you could use the relatively much easier-to-remember fraction 355/113 instead.

But how well can any irrational number like pi, e, phi, or √2 be approximated with a simple fraction with an integer numerator and denominator?

That has been an open question since 1941, when Richard Duffin and Albert Schaeffer conjectured that for whatever level of error you’re willing to live with in your rational approximation of an irrational number, you can either find a nearly infinite number of possible fractions or almost none. Here’s how Quanta Magazine‘s Kevin Hartnett described the conjecture:

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….

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.)

In what may be the biggest math story of 2019, the Duffin-Schaeffer conjecture may have been proven this past summer. Dimitris Koukoulopoulos and James Maynard posted a preprint of their paper confirming that choosing smaller ‘acceptable’ error ranges makes it harder to approximate irrational numbers with simple fractions, which would be a remarkable advance in the field of number theory.

Scientific American‘s Leila Sloman describes their approach:

Maynard and Koukoulopoulos knew that previous work in the field had reduced the problem to one about the prime factors of the denominators—the prime numbers that, when multiplied together, yield the denominator. Maynard suggested thinking about the problem as shading in numbers: “Imagine, on the number line, coloring in all the numbers close to fractions with denominator 100.” The Duffin-Schaeffer conjecture says if the errors are large enough and one does this for every possible denominator, almost every number will be colored in infinitely many times.

For any particular denominator, only part of the number line will be colored in. If mathematicians could show that for each denominator, sufficiently different areas were colored, they would ensure almost every number was colored. If they could also prove those sections were overlapping, they could conclude that happened many times. One way of capturing this idea of different-but-overlapping areas is to prove the regions colored by different denominators had nothing to do with one another—they were independent.

But this is not actually true, especially if two denominators share many prime factors. For example, the possible denominators 10 and 100 share factors 2 and 5—and the numbers that can be approximated by fractions of the form n/10 exhibit frustrating overlaps with those that can be approximated by fractions n/100.

Maynard and Koukoulopoulos solved this conundrum by reframing the problem in terms of networks that mathematicians call graphs—a bunch of dots, with some connected by lines (called edges). The dots in their graphs represented possible denominators that the researchers wanted to use for the approximating fraction, and two dots were connected by an edge if they had many prime factors in common. The graphs had a lot of edges precisely in cases where the allowed denominators had unwanted dependencies.

Using graphs allowed the two mathematicians to visualize the problem in a new way. “One of the biggest insights you need is to forget all the unimportant parts of the problem and to just home in on the one or two factors that make [it] very special,” says Maynard. Using graphs, he says, “not only lets you prove the result, but it’s really telling you something structural about what’s going on in the problem.” Maynard and Koukoulopoulos deduced that graphs with many edges corresponded to a particular, highly structured mathematical situation that they could analyze separately.

The graphs they develop to map the greatest common divisors for the rational approximations of irrational numbers are bipartite graphs. The following video provides a short introduction:

The Koukoulopoulos-Maynard proof of the Duffin-Schaeffer conjecture is now in the process of being validated. If determined to be valid, the proof may have an immediate impact on the related field of p-Adic approximation, which would have applications in quantum mechanics and field theory, as well as resolving other conjectures in number theory that rely on the Duffin-Schaeffer conjecture being true.

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