Disjunctive Omega-Words and Real Numbers
Peter Hertling (Theoretische Informatik I, FernUniversität Hagen, Germany)
Abstract: An ω-word p over a finite alphabet Σ is called disjunctive if every finite word over Σ occurs as a subword in p. A real number is called disjunctive to base a if it has a disjunctive a-adic expansion. For every pair of integers a,b ≥ 2 such that there exist numbers disjunctive to base a but not to base b we explicitly construct very simple examples of such numbers. General versions of the following results are proved. If (ni)i∈ω is a strictly increasing sequence of positive integers with ni+1 ≥ 3ni for infinitely many i then Σ 3-ni is disjunctive to base 2. The number Σ2-i!-i is disjunctive to base a if a is even and not a power of 2. The sum Σ2-ci is disjunctive to base 6 if c ≥ 3 is odd.
Keywords: disjunctiveness, invariant properties, normality, number representations, periods of rational numbers, ω-words