Totally real cubic numbers are well approximable
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.
Keywords
Cite
@article{arxiv.2310.12703,
title = {Totally real cubic numbers are well approximable},
author = {Alan Haynes},
journal= {arXiv preprint arXiv:2310.12703},
year = {2023}
}
Comments
Nikolay Moshchevitin pointed out an error in the last displayed equation: the claim that it is a subset of A_uR_T is not true. u_1 can fluctuate by a multiplicative factor of up to kappa. This forces the last part of the argument to sample a geometric subsequence of the unipotent flow. We acknowledge this is a non-trivial gap, and withdraw our claim of having solved this problem