Geodesic
ON THE SPECTRA OF PISOT NUMBERS
Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 127-132

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DOI

Let θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ N. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such thatfor allan ∈ {0,1}, N ∈ N}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.
DOI : 10.1017/S0017089511000462
Mots-clés : 11R80, 11J71, 11R06 
ZAIMI, TOUFIK. ON THE SPECTRA OF PISOT NUMBERS. Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 127-132. doi: 10.1017/S0017089511000462
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