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On the fractional parts of powers of algebraic numbers

In this chapter we focus on the sequences ({*ξ*α^{n}})_{n≥1} and (||*ξ*α^{n}||)_{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 ({*α*^{n}})_{n≥1} has at most two limit points, namely 0 and 1. For example, 0 and 1 are the limit points of the sequence while tends to 1 as *n* tends to infinity. The situation is rather different if *α* is a Salem number, since (*α*^{n})_{n≥1} is then dense modulo one, as was proved by Pisot ...