2

On the fractional parts of powers of real numbers

We established in the previous chapter several metrical statements on the distribution modulo one of sequences (*ξ**α*^{n})_{n≥1}. However, very little is known for *given* real numbers *ξ* and *α*. This chapter and the next one are mainly concerned with the following general questions.

(Hardy 1919) *Do there exist a transcendental real number α* > 1

(Mahler, 1968) *Given a real number α* > 1

The second of these questions was asked by Mahler ...

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