There is a word with precise mathematical meaning that matches naive non-mathematical intuition: "almost all" / "almost none" (aka full measure set).
This concept is easy to explain to lay people: count up lengths of intervals, do some limiting process trickery to deal with the fact that the set cannot be written as a finite union of intervals. Alternatively, choose a real number uniformly at random from some interval, probability one will hit the "almost all" set and probability zero will hit one of the others.
"negligible set" is often used to describe sets that have zero measure and small Baire category. "Virtually" also has different mathematical connotations.
Upside is that it was well explained that the contents of the new result is "don't worry about overlaps between different denominators".