Diophantine approximation in prescribed degree

TL;DR

This paper explores how well real numbers can be approximated by algebraic numbers of a fixed degree, providing new results on quadratic irrationals, Liouville numbers, and inhomogeneous approximation, with implications for Diophantine approximation theory.

Contribution

It answers a question by Bugeaud on quadratic approximation, refines a 1967 result by Davenport and Schmidt, and offers new characterizations of Liouville numbers and insights into inhomogeneous approximation.

Findings

Solved Bugeaud's approximation question for quadratic irrationals.

Refined Davenport and Schmidt's 1967 approximation bounds.

Provided new characterizations of Liouville numbers.

Abstract

We investigate approximation to a given real number by algebraic numbers and algebraic integers of prescribed degree. We deal with both best and uniform approximation, and highlight the similarities and differences compared with the intensely studied problem of approximation by algebraic numbers (and integers) of bounded degree. We establish the answer to a question of Bugeaud concerning approximation to transcendental real numbers by quadratic irrational numbers, and thereby we refine a result of Davenport and Schmidt from 1967. We also obtain several new characterizations of Liouville numbers, and certain new insights on inhomogeneous Diophantine approximation. As an auxiliary side result, we provide an upper bound for the number of certain linear combinations of two given relatively prime integer polynomials with a linear factor. We conclude with several open problems.