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Distribution Modulo One and Diophantine Approximation by Yann Bugeaud

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3

On the fractional parts of powers of algebraic numbers

In this chapter we focus on the sequences ({ξαn})n≥1 and (||ξαn||)n≥1, where ξ is a non-zero real number and α is a real algebraic number greater than 1. We first observe that, if α is a Pisot number, then the sequence ({αn})n≥1 has at most two limit points, namely 0 and 1. For example, 0 and 1 are the limit points of the sequence img while img tends to 1 as n tends to infinity. The situation is rather different if α is a Salem number, since (αn)n≥1 is then dense modulo one, as was proved by Pisot ...

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