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 in the particular case where α =3/2 and [s, s + t) = [0, 1/2).

Section 2.1 is devoted to classical results of Pisot and of Vijayaraghavan, which are also presented in Salem's monograph [619] and in [80]. In the next two sections, we investigate the set of pairs (ξ, α) for which the sequence ({ξαn})n≥1 avoids an interval of positive length included in [0, 1]. Among other results, we show in Section 2.3 that, however close to 1 the real number α > 1 can be, there always exist non-zero real numbers ξ such that the sequence ({ξαn})n≥1 enjoys the latter property.