Year: 2023
Type: preprint
Abstract: In this paper we prove that all irrational numbers from totally real cubic number fields are well approximable by rationals (i.e. the partial quotients in the continued fraction expansion of such a number are unbounded). This settles the long standing open question of whether or not well approximable algebraic numbers exist. Our proof uses a number theoretic classification of approximations to algebraic numbers, together with a result of Lindenstrauss and Weiss which is an application of Ratner's orbit closure theorem.
Source: arXiv (Cornell University)
Author: Haynes, Alan
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4387838960