On fractal measures and diophantine approximation

On Fractal Measures and Diophantine Approximation

We study diophantine properties of a typical point with respect to measures on Rn. Namely, we identify geometric conditions on a measure μ on Rn guaranteeing that μ-almost every y ∈ Rn is not very well multiplicatively approximable by rationals. Measures satisfying our conditions are called ‘friendly’. Examples include smooth measures on nondegenerate manifolds; thus this paper generalizes the ...

Metric Diophantine approximation and ‘ absolutely friendly ’ measures

Let W (ψ) denote the set of ψ-well approximable points in Rd and let K be a compact subset of Rd which supports a measure μ. In this short note, we show that if μ is an ‘absolutely friendly’ measure and a certain μ–volume sum converges then μ(W (ψ) ∩K) = 0. The result obtained is in some sense analogous to the convergence part of Khintchines classical theorem in the theory of metric Diophantine...

Complex Dimensions of Self-Similar Fractal Strings and Diophantine Approximation

Diophantine approximation and Diophantine equations

The first course is devoted to the basic setup of Diophantine approximation: we start with rational approximation to a single real number. Firstly, positive results tell us that a real number x has “good” rational approximation p/q, where “good” is when one compares |x − p/q| and q. We discuss Dirichlet’s result in 1842 (see [6] Course N◦2 §2.1) and the Markoff–Lagrange spectrum ([6] Course N◦1...

Diophantine Approximation on Veech

— We show that Y. Cheung’s general Z-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an ...