Restricted rational approximation and Apéry-type constructions

Effective Results for Restricted Rational Approximation to Quadratic Irrationals

In this paper, we deduce a number of effective lower bounds upon the distance to an integer of quantities of the shape bnξ, where b and n are integers and ξ is a real quadratic irrational. For certain ξ, we obtain inequalities that approach in strength the corresponding ineffective results arising from the p-adic version of Roth’s theorem.

Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions

Each non-zero point in Rd identifies one closest point x on the unit sphere Sd−1. We are interested in computing an ε-approximation y ∈ Qd for x , that is exactly on Sd−1 and has low bit size. We revise lower bounds on rational approximations and provide explicit, spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in R2 and R3. Mo...

Rational Approximation

Recently several authors have investigated the question of approximating certain functions by reciprocals of polynomials under the uniform norm on the positive real axis . Perhaps these results have some applications in industry and elsewhere (cf. [1, 36]) . Our present motivation is to give a detailed list of all the known results with simplified proofs in some cases and many new results and f...

fragmentability and approximation

in chapter 1, charactrizations of fragmentability, which are obtained by namioka (37), ribarska (45) and kenderov-moors (32), are given. also the connection between fragmentability and its variants and other topics in banach spaces such as analytic space, the radone-nikodym property, differentiability of convex functions, kadec renorming are discussed. in chapter 2, we use game characterization...

Restricted Nonlinear Approximation 1