Three of the main questions that motivate the present book are the following:

• Is there a transcendental real number α such that ∥αn∥ tends to 0 as n tends to infinity?

• Is the sequence of fractional parts {(3/2)n}, n ≥ 1, dense in the unit interval?

• What can be said on the digital expansion of an irrational algebraic number?

The latter question amounts to the study of the sequence (ξ10n)n≥1 modulo one, where ξ is an irrational algebraic number. More generally, for given real numbers ξ ≠ 0 and α > 1, we are interested in the distribution of the sequences ({ξαn})n≥1 and (∥ξαn∥)n≥1, where {·} (resp., ∥·∥) denotes the fractional part (resp., the distance to the nearest integer). The situation is very well understood from a metrical point of view. However, for a given pair (ξ, α), our knowledge on ({ξαn})n≥1 is extremely limited, except in very few cases. For instance when ξ = 1 and α is a Pisot number, that is, an algebraic integer (an algebraic integer is an algebraic number whose minimal defining polynomial over ℤ is monic) all of whose Galois conjugates (except itself) are lying in the open unit disc, it is not difficult to show that ∥αn∥ tends to 0 as n tends to infinity.