Simultaneous Approximation to Real and p-adic Numbers

TL;DR

This paper investigates the limits of simultaneously approximating real and p-adic numbers using roots of integer polynomials, improving existing bounds and constructing optimal examples, with implications for algebraic number approximation.

Contribution

It develops new constraints on approximation quality for real and p-adic numbers and extends existing results to algebraic numbers of degree 3 or 4 with fixed coefficients.

Findings

Established bounds on approximation improvements beyond Dirichlet's principle.

Constructed examples demonstrating the optimality of these bounds.

Extended Roy's work to show the golden ratio's square as optimal for degree 4 algebraic approximation.

Abstract

We study the problem of simultaneous approximation to a fixed family of real and p-adic numbers by roots of integer polynomials of restricted type. The method that we use for this purpose was developed by H. Davenport and W.M. Schmidt in their study of approximation to real numbers by algebraic integers. This method based on Mahler's Duality requires to study the dual problem of approximation to successive powers of these numbers by rational numbers with the same denominators. Dirichlet's Box Principle provides estimates for such approximations but one can do better. In this thesis we establish constraints on how much better one can do when dealing with the numbers and their squares. We also construct examples showing that at least in some instances these constraints are optimal. Going back to the original problem, we obtain estimates for simultaneous approximation to real and p-adic numbers by roots of integer polynomials of degree 3 or 4 with fixed coefficients in degree at least 3. In the case of a single real number (and no p-adic numbers), we extend work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace.