 
Summary: EXTREMAL PROPERTIES OF (EPI)STURMIAN SEQUENCES AND
DISTRIBUTION MODULO 1
JEANPAUL ALLOUCHE AND AMY GLEN
Abstract. Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey some combi
natorial properties of (epi)Sturmian sequences and distribution modulo 1 in connection to
their work. In particular we focus on extremal properties of (epi)Sturmian sequences, some
of which have been rediscovered several times.
1. Introduction
A little while ago, the first author came across a paper of Y. Bugeaud and A. Dubickas
[22] where the authors describe all irrational numbers > 0 such that the fractional parts
{bn}, n 0, all belong to an interval of length 1/b, where b 2 is a given integer. They
also prove that 1/b is minimal, i.e., for any interval I of length < 1/b, there is no irrational
number > 0, such that the fractional parts {bn}, n 0, all belong to I. An interesting
and unexpected result in their paper is the following: the irrational numbers > 0 such that
the fractional parts {bn}, n 0, all belong to a closed interval of length 1/b are exactly
the positive real numbers whose base b expansions are characteristic Sturmian sequences on
{k, k +1}, where k {0, 1, . . . , b2}. We recall that Sturmian sequences (resp. characteristic
Sturmian sequences) are the codings of trajectories on a square billiard that start from a side
(resp. from a corner) with an irrational slope; alternatively a Sturmian (resp. characteristic
