01404nas a2200157   4500008004100000022001400041245012700055210006900182300001000251490000800261520088500269653001301154100001501167700001301182856005101195       2012                        eng d  a0065-103600aAn equivalent of Kronecker's Theorem for powers of an Algebraic Number and Structure of Linear Recurrences of fixed length0 aequivalent of Kroneckers Theorem for powers of an Algebraic Numb  a15-330 v1533 a<p>After defining a notion of epsilon-density, we provide for any integer m&gt;1 and real algebraic number alpha an estimate of the smallest epsilon such that the set of vectors of the form (t,t^alpha,...,t alpha^{m-1}) for tR is epsilon-dense modulo 1 in terms of the multiplicative Mahler measure M(A(x)) of the minimal integral polynomial A(x) of alpha, which is independent of m. In particular, we show that if alpha has degree d it is possible to take epsilon = 2^{[d/2]}/M(A(x)). On the other side we show using asymptotic estimates for Toeplitz determinants that we cannot have epsilon$-density for sufficiently large m whenever epsilon is strictly smaller than 1/M(A(x)). In the process of proving this we obtain a result of independent interest about the structure of the Z-module of integral linear recurrences of fixed length determined by a non-monic polynomial.</p>
10aRobotics1 aDubbini, N1 aMonge, M  uhttp://journals.impan.pl/cgi-bin/doi?aa153-1-2