Totally real cubic numbers are well approximable

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.