Piecewise syndetic set

In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.
A set formula_1 is called "piecewise syndetic" if there exists a finite subset "G" of formula_2 such that for every finite subset "F" of formula_2 there exists an formula_4 such that
where formula_6. Equivalently, "S" is piecewise syndetic if there is a constant "b" such that there are arbitrarily long intervals of formula_2 where the gaps in "S" are bounded by "b".
Other notions of largeness.
There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers: