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.

A complex number α is said to satisfy the height reducing property if there is a finite set F ⊂ Z such that Z[α] = F [α], where Z is the ring of the rational integers. It is easy to see that α is an algebraic number when it satisfies the height reducing property. We prove the relation Card(F ) ≥ max{2, |Mα(0)|}, where Mα is the minimal polynomial of α over the field of the rational numbers, and discuss the related optimal cases, for some classes of algebraic numbers α. In addition, we show that there is an algorithm to determine the minimal height polynomial of a given algebraic number, provided it has no conjugate of modulus one.Card(F ) ≤ Card(S)(Card(S) s+1 − 1)/(Card(S) − 1), and, hence, J (t−s) (0) = β which implies that β ∈ U . Thus U ⊂ ℘, a contradiction, because by (9) we have that cZ ⊂ U and so U cannot be finite.

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α ∈ P the fractional parts of the numbers α n N , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N 2d − 4. We also show that for any α ∈ P the integer parts of the numbers α n are divisible by N for infinitely many n if and only if N 2d − 3.

A complex number α is said to satisfy the height reducing property if there is a finite subset, say F , of the ring Z of the rational integers such that. This property has been considered by several authors, especially in contexts related to self affine tilings and expansions of real numbers in non-integer bases. We prove that a number satisfying the height reducing property, is an algebraic number whose conjugates, over the field of the rationals, are all of modulus one, or all of modulus greater than one. Expecting the converse of the last statement is true, we show some theoretical and experimental results, which support this conjecture.

Let P (K ) be the set of Pisot numbers generating a real algebraic number field K over the field of rationals Q. Then, a result of Meyer implies that P (K ) is relatively dense in the interval [1, ∞) and a theorem of Pisot gives that P (K ) contains units, whenever K = Q.In the present note, we prove analogous results for the set of complex Pisot numbers generating a non-real number field K over Q when K is neither a quadratic field nor a CM-field.