In this paper we show that the Hausdorff dimension of the set of singular pairs is . We also show that the action of diag(e t , e t , e −2t ) on SL3R/SL3Z admits divergent trajectories that exit to infinity at arbitrarily slow prescribed rates, answering a question of A. N. Starkov. As a by-product of the analysis, we obtain a higher-dimensional generalization of the basic inequalities satisfied by convergents of continued fractions. As an illustration of the technique used to compute Hausdorff dimension, we reprove a result of I. J. Good asserting that the Hausdorff dimension of the set of real numbers with divergent partial quotients is 1 2.

We prove that the set of singular vectors in R d , d ≥ 2, has Hausdorff dimension d 2 d+1 and that the Hausdorff dimension of the set of ε-Dirichlet improvable vectors in R d is roughly d 2 d+1 plus a power of ε between d 2 and d. As a corollary, the set of divergent trajectories of the flow by diag(e t , . . . , e t , e −dt ) acting on SL d+1 (R)/ SL d+1 (Z) has Hausdorff codimension d d+1 . These results extend the work of the first author in [6].

Abstract. We construct a Poincaré section for the horocycle flow on the modular surface SL(2, R)/SL(2, Z), and study the associated first return map, which coincides with a transformation (the BCZ map) defined by Boca-Cobeli-Zaharescu [8]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.

45 pages, 2 figures. Added motivational material requested by referee, in particular, a detailed sketch of the proof of main theorem was added to introduction. The notion of Z-expansion/Liouville direction was added as in order to abstract the proof of the Hausdorff dimension zero result.International audienceWe consider billiards in a (1/2)-by-1 rectangle with a barrier midway along a vertical side. Let NE be the set of directions theta such that the flow in direction theta is not ergodic. We show that the Hausdorff dimension of the set NE is either 0 or 1/2, with the latter occurring if and only if the length of the barrier satisfies the condition of P'erez Marco, i.e. the sum of (loglog q_{k+1})/q_k is finite, where q_k is the the denominator of the kth convergent of the length of the barrier

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...