Moscow Mathematical Journal

Volume 18, Issue 3, July–September 2018 pp. 491–516.

Diophantine Approximation in Prescribed Degree

Authors: Johannes Schleischitz (1)
Author institution:(1) Department of Mathematics and Statistics, University of Ottawa, Canada

Summary:

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.

2010 Math. Subj. Class. 11J13, 11J82, 11R09.

Keywords: Exponents of Diophantine approximation, Wirsing's problem, geometry of numbers, continued fractions.

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