Seminars & Colloquia Calendar
Spacings of Fractions/Torsion Points
Matthew Welsh, Rutgers University
Date & time: Friday, 02 February 2018 at 1:40PM - 2:40PM
|Abstract: It's a basic fact that rational numbers are dense in the real line, and quantifying this density in terms of the size of the denominators of the fractions is the content of Dirichlet's approximation theorem. We'll frame our discussion of the spacings between fractions through the question: is Dirichlet's theorem (more or less) optimal? The answer is yes, and for a quite easy reason. But depending on how the room feels, we can make plenty of digressions that highlight some basic techniques in analytic number theory. For example, the equidistribution of fractions, its limitations, and its connection to the Riemann hypothesis. With any remaining time, we can discuss some new, but still very elementary, work on the spacings between torsion points (pairs of fractions with the same denominator), which (kind of) answers the question: is the simultaneous version of Dirichlet's approximation theorem optimal?