In this paper we prove the Hausdorff dimension of the set of (nondegenerate) singular two-dimensional vectors with uniform exponent µ ∈ (1/2, 1) is 2(1 − µ) when µ ≥ √ 2/2, whereas for µ < √ 2/2 it is greater than 2(1 − µ) and at most (3 − 2µ)(1 − µ)/(1 + µ + µ 2 ). We also establish that this dimension tends to 4/3 (which is the dimension of the set of singular two-dimensional vectors) when µ tends to 1/2. These results improve upon previous estimates of R. Baker, joint work of the first author with M. Laurent, and unpublished work of M. Laurent. We also prove a lower bound on the packing dimension that is strictly greater than the Hausdorff dimension for µ ≥ 0.565 . . . . Definition 1. Let m, n be positive integers and A a real n × m matrix. The matrix A is badly approximable if there exists a positive constant c such that the system of inequalitieshas no solution x in Z m for any X ≥ 1.Definition 2. Let m, n be positive integers and A a real n × m matrix. We say that Dirichlet's Theorem can be improved for the matrix A if there exists a positive constant c < 1 such that the system of inequalities (1.1) has a solution x in Z m for any sufficiently large X.If the subgroup G = AZ m + Z n of R n generated by the m rows of the matrix t A (here and below, t M denotes the transpose of a matrix M) together with Z n has rank strictly less than m + n, then there exists x in Z m with |x| ∞ arbitrarily large, such that Ax = 0 and, consequently, for any real number w and any sufficiently large X > 1, the system of inequalitieshas a solution x in Z m . In several of the questions considered below, we have to exclude this degenerate situation, thus we are led to introduce the set M * n,m (R) of n × m matrices for which the associated subgroup G has rank m + n.When m = n = 1, that is, when A = (ξ) for some irrational real number ξ, it is not difficult to show that Dirichlet's Theorem can be improved if, and only if, ξ is badly approximable (or, equivalently, ξ has bounded partial quotients in its continued fraction expansion); see [19] and [13] for a precise statement. Furthermore, by using the theory of continued fractions, one can prove that, for any irrational real number ξ, there are arbitrarily large integers X such that the system of inequalities xξ|| ≤ 1 2X and 0 < x ≤ X (1.2) has no integer solutions; see Proposition 2.2.4 of [5].Since the set of badly approximable numbers has Lebesgue measure zero and Hausdorff dimension 1, this implies that the set of 1 × 1 matrices A for which Dirichlet's Theorem can be improved has Lebesgue measure zero and Hausdorff dimension 1. The latter assertion has been extended as follows.Theorem A. For any positive integers m, n, the set of real n×m matrices for which Dirichlet's Theorem can be improved has mn-dimensional Lebesgue measure zero and Hausdorff dimension mn.The first assertion of Theorem A has been established by Davenport and Schmidt [14] when min{m, n} = 1. According to Kleinbock and Weiss [21], their proof can be generalized to n × m matrices. Actually, a more general re...
