James Maynard (University of Oxford)

Approximating Real Numbers by Fractions

How well can you approximate real numbers by rationals 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! James talks about this problem and presents an approach that relies on a fun blend of different ideas, including ergodic theory, analytic number theory and graph theory.