Abstract. We prove that the countable intersection of C 1 -diffeomorphic images of certain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in R d , improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidt's (α, β)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automatically to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs.

Given an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

We investigate the question of how well points on a nondegenerate k k -dimensional submanifold M ⊆ R d M \subseteq \mathbb R^d can be approximated by rationals also lying on M M , establishing an upper bound on the “intrinsic Dirichlet exponent” for M M . We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of k k and d d which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on M M distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the simplex lemma for rationals lying on M M which provides new insights on the local distribution of rational points on nondegenerate manifolds.

We construct (α, β) and α-winning sets in the sense of Schmidt's game, played on the support of certain measures (absolutely friendly) and show how to compute the Hausdorff dimension for some.In particular, we prove that if K is the attractor of an irreducible finite family of contracting similarity maps of R N satisfying the open set condition, (the Cantor's ternary set, Koch's curve and Sierpinski's gasket to name a few known examples), then for any countable collection of nonsingular affine transformations, Λ i :where BA is the set of badly approximable vectors in R N .

Abstract. We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular m × n matrices are both equal to mn 1 − 1 m+n , thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt.Résumé. Nousétablissons une nouvelle connexion entre l'approximation métrique diophantine et la géométrie paramétrique des nombres en prouvant un principe variationnel facilitant le calcul des dimensions d'Hausdorff et de packing de nombreux ensembles d'intérêt dans l'approximation diophantienne. Nous montrons que les dimensions précitées de l'ensemble des matrices m × n singulières sont toutes deuxégales a mn 1 − 1 m+n , démontrant ainsi une conjecture de Kadyrov, Kleinbock, Lindenstrauss, et Margulis et répondantà une question de Bugeaud, Cheung, et Chevallier. D'autres applications comprennent le calcul des dimensions des ensembles des points témoignant des conjectures de Starkov et de Schmidt. Main resultsThe notion of singularity (in the sense of Diophantine approximation) was introduced by Khintchine, first in 1937 in the setting of simultaneous approximation [11], and later in 1948 in the more general setting of matrix approximation [12]. 1 Since then this notion has been studied within Diophantine approximation and allied fields, see Moshchevitin's 2010 survey [13]. An m × n matrix A is called singular if for all ε > 0, there exists Q ε such that for all Q ≥ Q ε , there exist integer vectors p ∈ Z m and q ∈ Z n such that Aq + p ≤ εQ −n/m and 0 < q ≤ Q.Here · denotes an arbitrary norm on R m or R n . We denote the set of singular m × n matrices by Sing(m, n). For 1 × 1 matrices (i.e. numbers), being singular is equivalent to being rational, and in general any matrix A which satisfies an equation of the form Aq = p, with p, q integral and q nonzero, is singular. However, Khintchine proved that there exist singular 2 × 1 matrices whose entries are linearly independent over Q [10, Satz II], and his argument generalizes to the setting of m×n matrices for all (m, n) = (1, 1). The name singular derives from the fact that Sing(m, n) is a Lebesgue nullset for all m, n, see e.g. [11, p.431] or [2, Chapter 5,§7]. Note that singularity is a strengthening of the property of Dirichlet improvability introduced by Davenport and Schmidt [6].In contrast to the measure zero result mentioned above, the computation of the Hausdorff dimension of Sing(m, n) has been a challenge that so far only met with partial progress. The first breakthrough was made in 2011 by Cheung [3], who proved that the Hausdorff dimension of Sing(2, 1) is 4/3; this was extended in 2...