Wednesday, September 21
Milner 317, 12:30 PM
Title: Simultaneous rational approximations to the cube roots of 2 and 4, and more
Abstract: Let r be the cubed root of 2, and consider what happens when you try to approximate both r and r^2 by a pair of rationals of the form p1/q, p2/q. You must miss. Your error vector can be made arbitrarily small, but only at the price of taking q large. If you scale up your error vector by times q^(3/2), the error vector cannot be arbitrarily small. On the other hand, by the Dirichlet box principle there are choices of q which make it not too large.
Plot the error vectors, scaled up in this way. They lie approximately on nested homothetic copies of the ellipse r^2x^2 - rxy + y^2 = 18r. The points lie densely on each of the ellipses, and on each ellipse the distribution is in a sense asymptotically uniform.
We explain this and generalize.
Also, see Detailed Abstract (PS format).