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We will present a quantitative estimate on the failure of Kronecker's density theorem for the subgroup of the torus generated by the vector formed by m powers of an algebraic number, when m is big. We prove that the resulting subgroup is epsilon-dense, where epsilon 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, for m big enough; this optimality is proved by means of a result on linear recurrences of finite 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. |
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