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Archiv der Mathematik

, Volume 87, Issue 2, pp 124–128 | Cite as

On integer and fractional parts of powers of Salem numbers

  • Toufik Zaimi
Original Paper

Abstract.

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 \(\frac{{\alpha ^n }}{N}\), 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.

Mathematics Subject Classification (2000).

11R80 11J71 11A41 

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Copyright information

© Birkhäuser Verlag, Basel 2006

Authors and Affiliations

  1. 1.College of Sciences, Department of MathematicsKing Saud UniversityRiyadhSaudi Arabia

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