Cdmtcs Research Report Series Disjunctive Sequences: an Overview

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. 1 Notation and Definitions Let IN be the set of positive integers. The number of elements of a finite set S is denoted by card(S). For any finite set (alphabet) X let X∗ denote the free monoid of words (including the empty word 2) over X, and X the set of (infinite) sequences over X. Words on X are denoted by u, v, w; sequences over X are denoted by x,y, z. For W ⊆ X∗ the submonoid generated by W is denoted W ∗, and W ω is the set of infinite sequences formed by concatenating members of W ; finally, let X∞ = X∗∪Xω. A subset W ⊆ X∗ is called a language; an ω-language is a subset of X. For w ∈ X∗ and γ ∈ X∞ the concatenation of w and γ is written wγ. This defines in an obvious way a product WΓ of sets W ⊆ X∗ and Γ ⊆ X∞: WΓ = {wγ | w ∈ W,γ ∈ Γ}. For the sake of brevity we shall write wB, w∗ and w instead of {w}B, {w}∗ and {w}, respectively. By |w| we denote the length of the word w ∈ X∗. The set of all initial words (prefixes) of γ ∈ X∞ is A(γ) = {w ∈ X∗ | ∃γ′ ∈ X∞ wγ′ = γ}. ∗The first author has has been partially supported by Auckland University Research Grant, A18/XXXXX/62090/F3414050. †Computer Science Department, The University of Auckland, Private Bag 92109, Auckland, New Zealand; email: [email protected]. ‡Universität Koblenz-Landau, Fachbereich Informatik, Rheinau 1, D-56075 Koblenz, Germany, email: [email protected]. §Martin-Luther-Universität Halle-Wittenberg, Institut für Informatik, Kurt-Mothes-Str. 1, D-06120 Halle (Saale), Germany, email: [email protected]. The set of all subwords (factors, infixes) of γ ∈ X∞ is T(γ) = {w ∈ X∗ | ∃v ∈ X∗ ∃γ′ ∈ X∞ vwγ′ = γ}, and the set of all suffixes of γ ∈ X∞ is S(γ) = {γ′ ∈ X∞ | ∃w ∈ X∗, wγ′ = γ}. For B ⊆ X∞ put