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Distribution Modulo One and Diophantine Approximation by Yann Bugeaud

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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 (ξαn)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 and a non-zero real number ξ such that ||ξαn|| tends to 0 as n tends to infinity?

(Mahler, 1968) Given a real number α > 1 and an interval [s, s + t) included in [0, 1), is there a non-zero real number ξ such that s ≤ {ξαn} < s + t for all integers n ≥ 0 ? What is the smallest possible t for which such a ξ does exist?

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

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