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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
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