There are infinitely many limit points of the fractional parts of powers

Suppose that α > 1 is an algebraic number and ξ > 0 is a real number. We prove that the sequence of fractional partsξα ⁿ , n = 1, 2, 3, …, has infinitely many limit points except when α is a PV-number and ξ ∈ ℚ(α). For ξ = 1 and α being a rational non-integer number, this result was proved by Vijayaraghavan.     

On the average difference between two conjugates of an algebraic number

We give a lower bound for |α−1|, where α is an algebraic number, and also an upper bound for the number of real zeros of a polynomial. A lower bound for the maximal modulus of conjugates of a totally real algebraic integer is also obtained. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 4, pp. 421–431, October–December, 1995.     

Mahler measures in a cubic field

We prove that every cyclic cubic extension E of the field of rational numbers contains algebraic numbers which are Mahler measures but not the Mahler measures of algebraic numbers lying in E. This extends the result of Schinzel who proved the same statement for every real quadratic field E. A corresponding conjecture is made for an arbitrary non-totally complex field E and some numerical examples are given. We also show that every natural power of a Mahler measure is a Mahler measure.     

On the fractional parts of the natural powers of a fixed number

Let ξ ≠ = 0 and α > 1 be reals. We prove that the fractional parts {ξ αⁿ}, n = 1, 2, 3, ..., take every value only finitely many times except for the case when α is the root of an integer: α = q ^1/d, where q ≥ 2 and d ≥ 1 are integers and ξ is a rational factor of a nonnegative integer power of α.     

On the maximal conjugate of a totally real algebraic integer

Let α be an algebraic integer of degreed whosed conjugates are all real. We give a lower bound for the absolute value of conjugates of α in terms ofd and of the number of conjugates outside the interval [−2; 2]. Combining this with a lower bound for Mahler's measure of a polynomial, we obtain a lower bound for the maximal conjugate of a totally real algebraic integer. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 1, pp. 18–25, January–March, 1997.     

Arithmetical Properties of Powers of Algebraic Numbers

We consider the sequences of fractional parts {ⁿ}, n = 1, 2,3,..., and of integer parts [ⁿ], n = 1, 2, 3,..., where isan arbitrary positive number and > 1 is an algebraic number.We obtain an inequality for the difference between the largestand the smallest limit points of the first sequence. Such aninequality was earlier known for rational only. It is alsoshown that for roots of some irreducible trinomials the sequenceof integer parts contains infinitely many numbers divisibleby either 2 or 3. This is proved, for instance, for , n = 1, 2, 3,.... The fact that thereare infinitely many composite numbers in the sequence of integerparts of powers was proved earlier for Pisot numbers, Salemnumbers and the three rational numbers 3/2, 4/3, 5/4, but nosuch algebraic number having several conjugates outside theunit circle was known. 2000 Mathematics Subject Classification11J71, 11R04, 11R06, 11A41.