## Arithmetical properties of powers of algebraic numbers.(English)Zbl 1164.11025

Summary: We consider the sequences of fractional parts $$\{\xi\alpha^n\}$$, $$n = 1, 2, 3,\dots,$$ and of integer parts $$[\xi\alpha^n]$$, $$n = 1, 2, 3,...,$$ where $$\xi$$ is an arbitrary positive number and $$\alpha > 1$$ is an algebraic number. We obtain an inequality for the difference between the largest and the smallest limit points of the first sequence. Such an inequality was earlier known for rational $$\alpha$$ only. It is also shown that for roots of some irreducible trinomials the sequence of integer parts contains infinitely many numbers divisible by either 2 or 3. This is proved, for instance, for $$[\xi((\sqrt{13}-1)/2)^n]$$, $$n = 1, 2, 3,....$$ The fact that there are infinitely many composite numbers in the sequence of integer parts of powers was proved earlier for Pisot numbers, Salem numbers and the three rational numbers 3/2, 4/3, 5/4, but no such algebraic number having several conjugates outside the unit circle was known.

### MSC:

 11J71 Distribution modulo one 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Full Text: