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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 multi-plicatively approximable by rationals. Measures satisfying our conditions are called 'friendly'. Examples include smooth measures on nondegenerate(More)
In this paper we present some results and applications of a new invariant for dynamical systems that can be viewed as a dy-namical analogue of topological dimension. This invariant has been introduced by M. Gromov, and enables one to assign a meaningful quantity to dynamical systems of innnite topological dimension and entropy. We also develop an(More)
Given ergodic p-invariant measures f i g on the 1-torus T = R=Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution 1 n converges to log p. We also prove a variant of this result for joinings of full entropy on T N. In conjunction with a method of Host, this yields the following. Denote q (x) = qx (mod 1). Then(More)
Let G be a split adjoint semisimple group over Q and K∞ ⊂ G(R) a maximal compact subgroup. We shall give a uniform, short and essentially elementary proof of the Weyl law for cusp forms on congruence quotients of G(R)/K∞. This proves a conjecture of Sarnak for Q-split groups, previously known only for the case G = PGL(n). The key idea amounts to a new type(More)