New York Journal of Mathematics
Volume 7 (2001) 303-318


Doug Hensley

The Geometry of Badly Approximable Vectors

Published: December 6, 2001
Keywords: badly approximable vector, distribution mod 1, translates of a lattice
Subject: 11J71 primary; 11K31, 11K36, 11K38, 11J13, 11J69, 11A55 and 11H99, secondary

A vector v=(v1,v2,..., vk) in Rk is ε-badly approximable if for all m, and for 1≦ j≦ k, the distance ||mvj|| from mvj to the nearest integer satisfies ||mvj||>ε m-1/k. A badly approximable vector is a vector that is ε-badly approximable for some ε>0. For the case of k=1, these are just the badly approximable numbers, that is, the ones with a continued fraction expansion for which the partial quotients are bounded. One main result is that if v is a badly approximable vector in Rk then as x→∞ there is a lattice Λ(v,x), said lattice not too terribly far from cubic, so that most of the multiples kv mod 1, 1≦ k≦ x, of v fall into one of O(x1/(k+1)) translates of Λ(v,x). Each translate of this lattice has on the order of xk/(k+1) of these elements. The lattice has a basis in which the basis vectors each have length comparable to x-1/(k+1), and can be listed in order so that the angle between each, and the subspace spanned by those prior to it in the list, is bounded below by a constant, so that the determinant of Λ(v,x) is comparable to x-k/(k+1).

A second main result is that given a badly approximable vector v=(v1,v2,..., vk), for all sufficiently large x there exist integer vectors nj,1≦ j≦ k+1∈ Zk+1 with euclidean norms comparable to x, so that the angle, between each nj and the span of the ni with i<j, is comparable to x-1-1/k, and the angle between (v1,v2,..., vk,1) and each nj is likewise comparable to x-1-1/k. The determinant of of the matrix with rows nj,1≦ j≦ k+1 is bounded. This is analogous to what is known for badly approximable numbers α but for the case k=1 we can arrange that the determinant be always 1.

Author information

Department of Mathematics, Texas A&M University, College Station, TX 77843-3368