Rich ω-words and monadic second-order arithmetic

Abstract

Rich ω-words are one-sided infinite strings which have every finite word as a subword (infix). Infix-regular w-words are one-sided infinite strings for which the infix set of a suffix is a regular language. We show that for a regular ω-language F (a set of predicates definable in Buchi's restricted monadic second order arithmetic) the following conditions are equivalent: 1. F contains a rich ω-word. 2. F is of second Baire category in the Cantor space of ω-words. 3. F is a non-nullset for a class of measures (including the natural Lebesgue measure on Cantor space). 4. F has maximum Hausdorff dimension.