There are infinitely many limit points of the fractional parts of powers

ArtŪras Dubickas

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There are infinitely many limit points of the fractional parts of powers

ArtŪras Dubickas

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Abstract

Suppose that $\al>1$ is an algebraic number and $\xi>0$ is a real number. We prove that the sequence of fractional parts $\{\xi \al^n\},$ $n =1,2,3,...,$ has infinitely many limit points except when $\al$ is a PV-number and $\xi \in \Q(\al).$ For $\xi=1$ and $\al$ being a rational non-integer number, this result was proved by Vijayaraghavan.

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ArtŪras Dubickas. 2005-12-14. There are infinitely many limit points of the fractional parts of powers. https://arxiv.org/abs/math/0512314

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