Missing digits and good approximations
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- by Andrew Granville;
- Bull. Amer. Math. Soc. 61 (2024), 23-53
- DOI: https://doi.org/10.1090/bull/1811
- Published electronically: October 16, 2023
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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 our 2015 Bulletin of the AMS article). 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 his 2019 partial resolution of 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 Schaeffer proposed an appropriate but significantly more general analogy involving approximation only by reduced fractions (which is much more useful). We will discuss its 2020 resolution by Maynard and Dimitris Koukoulopoulos.
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Bibliographic Information
- Andrew Granville
- Affiliation: Départment de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7, Canada
- MR Author ID: 76180
- ORCID: 0000-0001-8088-1247
- Email: andrew.granville@umontreal.ca
- Received by editor(s): July 15, 2023
- Published electronically: October 16, 2023
- Additional Notes: The author was partially supported by NSERC of Canada, both by a Discovery Grant and by a CRC
- © Copyright 2023 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 61 (2024), 23-53
- MSC (2020): Primary 11J83, 11N05; Secondary 11A41, 11A63, 05C40, 11N32, 11N35
- DOI: https://doi.org/10.1090/bull/1811
- MathSciNet review: 4678570