A vector v=(v1,v2,..., vk) in Rk isε-badly approximable if for all m, and for1≦ j≦ k, the distance ||mvj|| from mvj to the nearestinteger satisfies ||mvj||ε m-1/k. A badlyapproximable vector is a vector that is ε-badlyapproximable for some ε0. For the case of k=1, theseare just the badly approximable numbers, that is, the ones with acontinued fraction expansion for which the partial quotients arebounded. One main result is that if v is a badlyapproximable vector in Rk then as x→∞ thereis a lattice Λ(v,x), said lattice not too terriblyfar from cubic, so that most of the multiples kv mod 1, 1≦k≦ x, of v fall into one of O(x1/(k+1)) translatesof Λ(v,x). Each translate of this lattice has on theorder of xk/(k+1) of these elements. The lattice has a basisin which the basis vectors each have length comparable tox-1/(k+1), and can be listed in order so that the anglebetween each, and the subspace spanned by those prior to it in thelist, is bounded below by a constant, so that the determinant ofΛ(v,x) is comparable to x-k/(k+1). A second main result is that given a badly approximable vectorv=(v1,v2,..., vk), for all sufficiently large x thereexist integer vectors nj,1≦ j≦ k+1∈ Zk+1 witheuclidean norms comparable to x, so that the angle, between eachnj and the span of the ni with ij, iscomparable to x-1-1/k, and the angle between (v1,v2,...,vk,1) and each nj is likewise comparable tox-1-1/k. The determinant of of the matrix with rowsnj,1≦ j≦ k+1 is bounded. This is analogous to what isknown for badly approximable numbers α but for the casek=1 we can arrange that the determinant be always 1.
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