Abstract
Let ξ ≠ = 0 and α > 1 be reals. We prove that the fractional parts {ξ αn}, n = 1, 2, 3, ..., take every value only finitely many times except for the case when α is the root of an integer: α = q 1/d, where q ≥ 2 and d ≥ 1 are integers and ξ is a rational factor of a nonnegative integer power of α.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 5, pp. 1071–1075, September–October, 2006.
Original Russian Text Copyright © 2006 Dubickas A.
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Dubickas, A. On the fractional parts of the natural powers of a fixed number. Sib Math J 47, 879–882 (2006). https://doi.org/10.1007/s11202-006-0096-4
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DOI: https://doi.org/10.1007/s11202-006-0096-4