Primes with missing digits
Appears in collections : Jean-Morlet Chair: Ergodic theory and its connections with arithmetic and combinatorics / Chaire Jean Morlet : Théorie ergodique et ses connexions avec l'arithmétique et la combinatoire, Exposés de recherche, Fields medallists - 2022
We will talk about recent work showing there are infinitely many primes with no $7$ in their decimal expansion. (And similarly with $7$ replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically vey difficult. The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.
