A vector v=(v₁,v₂,..., v_k) in R^k is ε-badly 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 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 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=(v₁,v₂,..., 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^-1-1/k, and the angle between (v₁,v₂,..., v_k,1) and each n_j is likewise comparable to x^-1-1/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.