Abstract:
Following Jürgensen and Thierrin [21] we say that an infinite sequence is disjunctive
if it contains any (finite) word, or, equivalently, if any word appears in the
sequence infinitely many times. “Disjunctivity” is a natural qualitative property; it is
weaker, than the property of “normality” (introduced by Borel [1]; see, for instance,
Kuipers, Niederreiter [24]). The aim of this paper is to survey some basic results
on disjunctive sequences and to explore their role in various areas of mathematics
(e.g. in automata-theoretic studies of ω-languages or number theory). To achieve
our goal we will use various instruments borrowed from topology, measure-theory,
probability theory, number theory, automata and formal languages.