On a question of G. Kuba
Let \(\beta\) be a real algebraic number greater than or equal to 1, and let \(\{x\}\) denote the fractional part of \(x.\) Set \(||x||=\min \big(\{x\}, 1-\{x\}\big).\) By a well-known theorem of Pisot and Vijayaraghavan, \(\lim _{n \to \infty} ||\beta^n||= 0\) if and only if \(\beta\) is a Pisot number. \textit{G. Kuba} [Arch. Math. 69, 156-163 (1997; Zbl 0899.11050)]) asked whether there are algebraic numbers \(\beta \geq 1,\) other than positive integers, for which \(\lim _{n \to \infty }\{\beta^n\}=0.\) The author shows that the answer to this question is negative. This was also done by the reviewer [Publ. Math. 56, 141-144 (2000; Zbl 0999.11035)].
scientific article; zbMATH DE number 1462956