On the failure of Kronecker's density theorem for powers of an algebraic number

Maurizio Monge (UFRJ)

We will present a quantitative estimate on the failure of Kronecker's density theorem for the ow in the torus generated by a vector formed by m powers of an algebraic number, when m is big. We prove that the resulting subgroup is -dense, where  is related to the Mahler measure of the algebraic number. The problem is motivated by a problem in control theory, where we assume that only the integral part of the behaviour is known. The estimate on the density is proved to be best-possible up to a constant, form big enough; this optimality is proved by means of a result on linear recurrences of nite length, and estimates on the determinant of Toeplitz matrices. We formulate a conjecture on the constant provinding the best possible estimate, relating our problem to algebraic dynamical systems on the torus. (joint work with N. Dubbini).