Abstract. We study diophantine properties of a typical point with respect to measures on R n . Namely, we identify geometric conditions on a measure µ on R n guaranteeing that µ-almost every y ∈ R n is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called 'friendly'. Examples include smooth measures on nondegenerate manifolds, thus the present paper generalizes the main result of [KM]. Another class of examples is given by measures supported on self-similar sets satisfying the open set condition, as well as their products and pushforwards by certain smooth maps.

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 m × n real matrix Y , an unbounded set T of parameters t = (t 1 , . . . , t m+n ) ∈ R m+n + with m i=1 t i = n j=1 t m+j and 0 < ε ≤ 1, we say that Dirichlet's Theorem can be ε-improved for Y along T if for every sufficiently large t ∈ T there are nonzero q ∈ Z n and p ∈ Z m such thatWe show that for any ε < 1 and any T 'drifting away from walls', see (1.8), Dirichlet's Theorem cannot be ε-improved along T for Lebesgue almost every Y . In the case m = 1 we also show that for a large class of measures µ (introduced in [KLW]) there is ε 0 > 0 such that for any unbounded T , any ε < ε 0 , and for µ-almost every Y , Dirichlet's Theorem cannot be ε-improved along T . These measures include natural measures on sufficiently regular smooth manifolds and fractals.Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space SL m+n (R)/ SL m+n (Z).

We show that the sets of weighted badly approximable vectors in R n are winning sets of certain games, which are modifications of (α, β)-games introduced by W. Schmidt in 1966. The latter winning property is stable with respect to countable intersections, and is shown to imply full Hausdorff dimension.

Abstract. For a large class of closed subsets C of R n , we show that the intersection of C with the set of badly approximable vectors has the same Hausdorff dimension as C. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor's ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson's open set condition.