Arithmetical properties of linear recurrent sequences

Let F(z)∈R[z] be a polynomial with positive leading coefficient, and let α>1 be an algebraic number. For r=degF>0, assuming that at least one coefficient of F lies outside the field Q(α) if α is a Pisot number, we prove that the difference between the largest and the smallest limit points of the sequence of fractional parts {F(n)αn}n=1,2,3,… is at least 1/ℓ(Pr+1), where ℓ stands for the so-called reduced length of a polynomial.