Approximating reals by rationals

Approximating reals by rationals

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James Maynard, University of Oxford

Register at: https://math.princeton.edu/minerva-2021

How well can you approximate real numbers by fractions with denominators coming from a given set? Although this old question has applications in many areas, in general this question seems impossibly hard - we don’t even know whether $e+\pi$ is rational or not!

If you allow for a tiny number of bad exceptions, then a beautiful dichotomy occurs - either almost everything can be approximated or almost nothing! I’ll talk about this problem and recent joint work with Dimitris Koukoulopoulos which classifies when these options occur, answering an old question of Duffin and Schaeffer. This relies on a fun blend of different ideas, including ergodic theory, combinatorics, analytic number theory and graph theory.