Generalisations of disjunctive sequences

The present paper proposes a generalisation of the notion of disjunctive (or rich) sequence, that is, of an infinite sequence of letters having each finite sequence as a subword. Our aim is to give a reasonable notion of disjunctiveness relative to a given set of sequences F . We show that a definition like “every subword which occurs at infinitely many different positions in sequences in F has to occur infinitely often in the sequence” fulfils properties similar to the original unrelativised notion of disjunctiveness. Finally, we investigate our concept of generalised disjunctiveness in spaces of Cantor expansions of reals.