Arithmetical properties of linear recurrent sequences (Q863963)

The author is interested in the range of the fractional part of \(P(n) \alpha^n\), where \(P\) is a polynomial of degree \(r\) with positive leading coefficient, and \(\alpha\) an algebraic number \(>1\). He proves that, provided either \(\alpha\) is non-Pisot or at least one of the coefficients of \(P\) does not belong to \(\mathbb Q (\alpha)\), the largest and the smallest-limit points of the sequence \(\{P(n) \alpha^n\}\) differ by at least \(1/\ell (p^{r+1})\). Here, \(\ell(F)\) is defined for a polynomial \(F\) by \(\ell(F):= \inf_GL(FG)\), where the infimum is taken over all real polynomials \(G\) with constant or leading coefficient equal to 1, and \(L\) applied to a polynomial is its length, i.e., the sum of the absolute values of its coefficients (\(\ell\) is called the ``reduced length''). The case where \(p(x)=x\) was addressed by the author [see Bull. Lond. Math. Soc. 38, No. 1, 70--80 (2006; Zbl 1164.11025)] and by \textit{L. Flatto, J. Lagarias} and \textit{A. D. Pollington} if furthermore \(\alpha\) is rational [see Acta Arith. 70, No. 2, 125--147 (1995; Zbl 0821.11038)]. Please finally note that reference [2] appeared: Proc. of Words 2005, S. Brlek, C. Reutenauer eds., Monographies du LACIM, 36, UQAM Montréal, Canada, 2005, pp. 57--64 and that the details for references [14] are Arch. Math., Brno 42, 151--158 (2006). Furthermore the details for reference [26] are \textit{A. Schinzel} [Funct. Approximatio, Comment. Math. 35, 271--306 (2006; Zbl 1192.12001)].