 
Summary: DISTRIBUTION MODULO 1 AND THE LEXICOGRAPHIC WORLD
JEANPAUL ALLOUCHE AND AMY GLEN
In honour of Paulo Ribenboim on the occasion of his 80th birthday
ABSTRACT. We give a complete description of the minimal intervals containing all fractional parts
{2n
}, for some positive real number , and for all n 0.
1. INTRODUCTION
In the paper [20] Mahler defined the set of Znumbers by
R, > 0, n 0, 0
3
2
n
<
1
2
where {z} is the fractional part of the real number z. Mahler proved that this set is at most countable.
It is still an open problem to prove that this set is actually empty. More generally, given a real number
> 1 and an interval (x, y) (0, 1) one can ask whether there exists > 0 such that, for all
n 0, we have x {n} < y (or the variant x {n} y). Flatto, Lagarias, and Pollington
[14, Theorem 1.4] proved that, if = p/q with p, q coprime integers and p > q 2, then any
