Simultaneous Diophantine approximation on the circle and Hausdorff dimension
نویسندگان
چکیده
منابع مشابه
Hausdorff dimension and Diophantine approximation
In the present survey paper, we explain how the theory of Hausdorff dimension and Hausdorff measure is used to answer certain questions in Diophantine approximation. The final section is devoted to a discussion around the Diophantine properties of the points lying in the middle third Cantor set.
متن کاملInhomogeneous Diophantine approximation on curves and Hausdorff dimension
The goal of this paper is to develop a coherent theory for inhomogeneous Diophantine approximation on curves in R akin to the well established homogeneous theory. More specifically, the measure theoretic results obtained generalize the fundamental homogeneous theorems of R.C. Baker (1978), Dodson, Dickinson (2000) and Beresnevich, Bernik, Kleinbock, Margulis (2002). In the case of planar curves...
متن کاملDiophantine approximation on manifolds and lower bounds for Hausdorff dimension
Given n ∈ N and τ > 1 n , let Sn(τ) denote the classical set of τ approximable points in R, which consists of x ∈ R that lie within distance q from the lattice 1 q Z for infinitely many q ∈ N. In pioneering work, Kleinbock & Margulis showed that for any non-degenerate submanifold M of R and any τ > 1 n almost all points on M are not τ -approximable. Numerous subsequent papers have been geared t...
متن کاملDiophantine Approximation, Khintchine's Theorem, Torus Geometry and Hausdorff Dimension
A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of ψ-approximable points in the plane. The construction and use of Haudsorff measure and di...
متن کاملSimultaneous Diophantine Approximation on Planar
Let C be a non-degenerate planar curve. We show that the curve is of Khintchine-type for convergence in the case of simultaneous approximation in R 2 with two independent approximation functions; that is if a certain sum converges then the set of all points (x, y) on the curve which satisfy simultaneously the inequalities qx < ψ1(q) and qy < ψ2(q) infinitely often has induced measure 0. This co...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Mathematical Proceedings of the Cambridge Philosophical Society
سال: 2001
ISSN: 0305-0041,1469-8064
DOI: 10.1017/s0305004101004984