Diophantine Properties of Powers of Some Pisot Numbers

摘要

Let α be a Pisot number, i.e., an algebraic integer greater than 1 whose conjugates lie strictly inside the unit disk D = {z ∈ C : |z| ＜ 1}. Let us denote the minimal polynomial for α by P(x) = x~d - a_1x~(d-1) - a_2x~(d-2) -…- a_kx~(d-k) -…- a_(d-1)x - a_d, where a_i ∈ Z. Let X = (x_n)_(n=1)~∞, x_n (→) 0, be a sequence satisfying the recurrence relation x_n = a_1x_(n-1) + a_2x_(n-2) +…+ a_kx_(n-k) +…+ a_(d-1)x_(n-d+1)+ a_dx_(n-d). (1) The geometric progression (α~n)_(n=1)~∞ is an example of such a sequence. In the present paper, we study the quantity L(X) := sup ξ∈R lim inf n→∞ ‖ξx_n‖. (Here and elsewhere, ‖ • ‖ denotes the distance to the nearest integer, while { • } is the fractional part and [•] is the integer part.)