An arithmetical property of powers of Salem numbers

Abstract

Let ζ be a nonzero real number and let α be a Salem number. We show that the difference between the largest and smallest limit points of the fractional parts of the numbers ζ α n , when n runs through the set of positive rational integers, can be bounded below by a positive constant depending only on α if and only if the algebraic integer α − 1 is a unit.

On integer and fractional parts of powers of Salem numbers

“…A similar inequality is proved in [9] when ζ ∈ Q(α), provided α − 1 is a unit. In fact the initial aim of the present work was to resolve the inequality (ζ, α) < 1, in the unknown ζ ∈ Q(α), when α − 1 is a unit.…”

“…it has been shown recently [9] that for any 0 < ε < 1 there is a Pisot number ζ ∈ Q(α) and a subinterval I ζ of ]0, 1[ with length ε such that L(ζ, α) ∩ I ζ = ∅. In [1], Dubickas has proved that L( 1 N , α) = [0, 1] when N ∈ {2, 3, 4}, and has deduced that the equations [α n ] = 0 mod N hold for infinitely many n ∈ N provided N ∈ {2, 3, 4}; he has also asked (implicitly in Remark 3) whether these later two properties hold for other values of N .…”

“…A proof of Lemma 1 can be found in [9]. This proof is based on Kronecker's theorem [5, appendix 8] and a result due to Pisot which says that the numbers…”

On the powers of 3/2 and other rational numbers

“…The subject of distribution of powers of a number modulo 1 is classical and goes back to [10], [11], [17][18][19]. There are, however, many recent publications [1][2][3][4][5][6][7], [20], [21] on this subject too. We will compare our results with some of these in the next section.…”

Linear recurrence sequences without zeros

“…The exceptions of the theorem proved in [4] are the pairs ξ, α, where α is a Pisot number or a Salem number and ξ lies in the field Q(α). The case of Salem numbers α and ξ ∈ Q(α) has been consider by Zaïmi in [21].…”