Approximation of p-adic numbers by algebraic numbers of bounded degree

Approximation of complex algebraic numbers by algebraic numbers of bounded degree

To measure how well a given complex number ξ can be approximated by algebraic numbers of degree at most n one may use the quantities w n (ξ) and w * n (ξ) introduced by Mahler and Koksma, respectively. The values of w n (ξ) and w * n (ξ) have been computed for real algebraic numbers ξ, but up to now not for complex, non-real algebraic numbers ξ. In this paper we compute w n (ξ), w * n (ξ) for a...

Approximation Lattices of p - adic Numbers

Approximation lattices occur in a natural way in the study of rational approximations to p-adic numbers. Periodicity of a sequence of approximation lattices is shown to occur for rational and quadratic p-adic numbers. and for those only, thus establishing a p-adic analogue of Lagrange’s theorem on periodic continued fractions. Using approximation lattices we derive upper and lower bounds for th...

On the Approximation to Algebraic Numbers by Algebraic Numbers

Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ−α| < H(α)−n−1+ε, where H(α) denotes the näıve height of α. We sharpen this result by replacing ε by a function H 7→ ε(H) that tends to zero wh...

Computable p–adic Numbers

In the present work the notion of the computable (primitive recursive, polynomially time computable) p–adic number is introduced and studied. Basic properties of these numbers and the set of indices representing them are established and it is proved that the above defined fields are p–adically closed. Using the notion of a notation system introduced by Y. Moschovakis an abstract characterizatio...

On approximation of real numbers by real algebraic numbers