 

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 


Abstract
A vector v=(v_{1},v_{2},..., v_{k}) in R^{k} is
εbadly approximable if for all m, and for
1≦ j≦ k, the distance mv_{j} from mv_{j} to the nearest
integer satisfies mv_{j}>ε m^{1/k}. A badly
approximable vector is a vector that is εbadly
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 R^{k} 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(x^{1/(k+1)}) translates
of Λ(v,x). Each translate of this lattice has on the
order of x^{k/(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=(v_{1},v_{2},..., v_{k}), for all sufficiently large x there
exist integer vectors n_{j},1≦ j≦ k+1∈ Z^{k+1} with
euclidean norms comparable to x, so that the angle, between each
n_{j} and the span of the n_{i} with i<j, is
comparable to x^{11/k}, and the angle between (v_{1},v_{2},...,
v_{k},1) and each n_{j} is likewise comparable to
x^{11/k}. The determinant of of the matrix with rows
n_{j},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 778433368
Doug.Hensley@math.tamu.edu
http://www.math.tamu.edu/~doug.hensley/

