Thursday, October 28
Milner 216, 1:00 PM
Title: Simultaneous diophantine approximation
Abstract: Simultaneous diophantine approximation of a target vector v=(v_1, ..., v_n) seeks a positive integer q, and an integer vector p=(p_1, ..., p_n) such that qv-p is in some sense short.
The Dirichlet box principle ensures that given Q>1, there exists q, 1 <= q <= Q such that (with optimal choice of p), |qv-p| << Q^{-1/n}. We define the scaled approximation error E(v,q) to be q^{1/n}(qv-p), where p is the integer vector nearest qv. Given some positive bound C, we call q a good denominator for v if the entries of this scaled error all have absolute value less than C.
When v takes the form v=(alpha, alpha^2, ..., alpha^n) where alpha is an algebraic integer of degree n+1, the scaled errors associated with good denominators for v are not distributed uniformly in the available box, but instead are plastered (nearly) onto a finite complex of Russian-doll surfaces.
Computational experiments suggest a similar result when the numerators and denominator are taken from the set of Gaussian integers.