Missing digits, and good approximations

Abstract

James Maynard has taken the analytic number theory world by storm in the last decade, proving several important and surprising theorems, resolving questions that had seemed far out of reach. He is perhaps best known for his work on small and large gaps between primes (which were discussed, hot off the press, in my 2015 CEB lecture). In this article we will discuss two other Maynard breakthroughs: -- Mersenne numbers take the form $2^n-1$ and so appear as $111\dots 111$ in base 2, having no digit `$0$'. It is a famous conjecture that there are infinitely many such primes. More generally it was, until Maynard's work, an open question as to whether there are infinitely many primes that miss any given digit, in any given base. We will discuss Maynard's beautiful ideas that went into partly resolving this question. -- In 1926, Khinchin gave remarkable conditions for when real numbers can usually be ``well approximated'' by infinitely many rationals. However Khinchin's theorem regarded 1/2, 2/4, 3/6 as distinct rationals and so could not be easily modified to cope, say, with approximations by fractions with prime denominators. In 1941 Duffin and Schaefer proposed an appropriate but significantly more general analogy involving approximation only by reduced fractions (which is much more useful). We will discuss its recent resolution by Maynard together with Dimitris Koukoulopoulos.