On the fractional parts of the natural powers of a fixed number

Arturas Dubickas

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On the fractional parts of the natural powers of a fixed number

机译：关于固定自然数的小数部分

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摘要

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.

机译：令ξ≠= 0且α> 1为实数。我们证明小数部分{ξαn}，n = 1，2，3，...，仅有限次地取每个值，除非当α是整数的根时：α= q 1 / d，其中q≥2和d≥1是整数，并且ξ是α的非负整数幂的有理因子。

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来源
《Siberian Mathematical Journal》 |2006年第5期|共4页
作者
Arturas Dubickas;
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原文格式 PDF
正文语种 eng
中图分类数学;
关键词
fractional part; algebraic integer; roots; power;

机译：小数部分;代数整数;根;幂;

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On the fractional parts of the natural powers of a fixed number

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