2023/10/19 by Alan Haynes, Haynes, Alan
Mathematics · #11J68 #37A10 #Dynamical Systems (math.DS) #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Analysis #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2310.12703
openalex publication_date 2023/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.