Hausdorff theory of dual approximation on planar curves

Inhomogeneous 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...

Asymptotic Approximation of Convex Curves; the Hausdorff Metric Case

where κ(t) is the curvature of C given as a function of the arclength t and l the length of C. See also McClure and Vitale [7] and for asymptotic formulae for approximation with respect to the Hausdorff metric in higher dimensions R. Schneider [8],[9] and P.M. Gruber [5]. In this article we extend the asymptotic formulae (1) by deriving the second terms in the asymptotic expansions of δ(C,P i n...

A note on simultaneous Diophantine approximation on planar curves

Let Sn(ψ1, . . . , ψn) denote the set of simultaneously (ψ1, . . . , ψn)–approximable points in Rn and S∗ n (ψ) denote the set of multiplicatively ψ–approximable points in Rn. Let M be a manifold in Rn. The aim is to develop a metric theory for the sets M∩Sn(ψ1, . . . , ψn) and M∩S∗ n (ψ) analogous to the classical theory in whichM is simply Rn. In this note, we mainly restrict our attention to...

Primitive Geometric Operations on Planar Algebraic Curves with Gaussian Approximation

We present a curve approximation method which approximates each planar algebraic curve segment by discrete curve points at each of which the curve has its gradient from a set of uniformly distributed normals. This method, called Gaussian Approximation (GAP), provides efficient algorithms for various primitive geometric operations, especially for those related with gradients such as common tange...

Primitive Geometri Operations on Planar Algebrai Curves with Gaussian Approximation