Singular vectors are those for which the quality of rational approximations provided by Dirichlet's Theorem can be improved by arbitrarily small multiplicative constants. We provide an upper bound on the Hausdorff dimension of singular vectors lying on self-similar fractals in R d satisfying the open set condition. The bound is in terms of quantities which are closely tied to Frostman exponents of projections of the Hausdorff measure supported on the fractal. Our bound is optimal in the sense that it agrees with the exact dimension of singular vectors obtained by Cheung and Chevallier when the fractal is trivial (i.e. has non-empty interior). As a corollary, we show that if the fractal is the product of 2 copies of Cantor's middle thirds set or the attractor of a planar homogeneous irrational IFS, then the upper bound is 2/3 the dimension of the fractal. This addresses the upper bound part of a question raised by Bugeaud, Cheung and Chevallier. We apply our method in the setting of translation flows on flat surfaces to show that the dimension of non-uniquely ergodic directions belonging to a fractal is at most 1/2 the dimension of the fractal.

We prove that for every flat surface \omega , the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from \omega exhibit a definite amount of deviation from the correct limit in Birkhoff’s and Oseledets’ theorems is strictly less than 1. This theorem extends a result by Chaika and Eskin who proved that such sets have measure 0. We also prove that the Hausdorff dimension of the directions in which Teichmüller geodesics diverge on average in a stratum is bounded above by 1/2, strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly-mixing IETs with permutation (d, d-1,\ldots,1) , where d \geq 5 is an odd number, is at least 1/2, thus strengthening a result by Avila and Leguil. Combined with a recent result of Chaika and Masur, this shows that the Hausdorff dimension of this set is exactly 1/2.

Let (X, B, µ) be a Borel probability space. Let T n : X → X be a sequence of continuous transformations on X. Let ν be a probability measure on X such that 2010 Mathematics Subject Classification. 37A30, and 22F30.

Suppose g t is a 1-parameter Ad-diagonalizable subgroup of a Lie group G and Γ < G is a lattice. We study differentiable curves of the form ϕ : [0, 1] → U + satisfying certain non-degeneracy conditions, where U + is the expanding horospherical subgroup of g t . For a class of examples that includes products of real rank one Lie groups, we obtain sharp upper bounds on the Hausdorff dimension of the set of points s for which the forward orbit (g t ϕ(s)x 0 ) t 0 is divergent on average in G/Γ, for any basepoint x 0 ∈ G/Γ. Moreover, we prove that the set of points s for which (g t ϕ(s)x 0 ) t 0 remains bounded is winning in the sense of Schmidt. We describe applications of our results to problems in diophantine approximation. Our methods also yield the following result for square systems of linear forms: suppose ϕ(s) = sY + Z where Y ∈ GL(n, R) and Z ∈ M n,n (R). Then, the dimension of the set of points s such that ϕ(s) is singular is at most 1/2 while badly approximable points have Hausdorff dimension equal to 1.Lemma 4.5. There exists a constant C 1 > 1, such that for all x ∈ X, natural numbers n with α(n) ≥ 1/γ, t > 0 and all subintervals J ⊂ [−1, 1] of radius at least e −α(nt) , one hasProof. First, we note that for all r ∈ [−1, 1], we haveUsing positivity of f , (4.5) and a change of variable, we getThen, Fubini's theorem and the fact that ϕ is C 1+γ imply the following. J f (g (n+1)t u(ϕ(s))x) ds J 1 −1 f (g (n+1)t u(ϕ(s) + re −α(nt)φ (s) + O(e −(1+γ)α(nt) ))x) dr dsMoreover, by definition of g t and u(Y ), we haveThus, by our assumption that α(n) 1/γ, we getIn particular, by (4.7), we getSince e α(t) ∈ N, for each 1 ≤ j ≤ k, the partition P k is a refinement of P j . This implies the following inclusion.Hence, the following inequality follows from (4.8), (4.9), and the fact that f is non-negative.(4.10) Iterating (4.10), by induction, we obtain the following exponential decay integral estimate. B(M,n−1)∩J 0 f (g (m+n)t u(ϕ(s))x) ds (2C 1c ) n e −βα(nt) J∈Pm J∩B(M,n−1)∩J 0 =∅ J f (g mt u(ϕ(s))x) ds = (2C 1c ) n e −βα(nt) J 0 f (g mt u(ϕ(s))x) ds (4.11)where on the second line, we used the following consequence of P m being a partition.

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