Piecewise syndetic set

In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.

A set $S\subset \mathbb {N}$ is called piecewise syndetic if there exists a finite subset G of $\mathbb {N}$ such that for every finite subset F of $\mathbb {N}$ there exists an $x\in \mathbb {N}$ such that

x+F\subset \bigcup _{n\in G}(S-n)

where $S-n=\{m\in \mathbb {N} :m+n\in S\}$ . Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of $\mathbb {N}$ where the gaps in S are bounded by b.

Properties

A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set.
If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.
A set S is piecewise syndetic if and only if there exists some ultrafilter U which contains S and U is in the smallest two-sided ideal of $\beta \mathbb {N}$ , the Stone–Čech compactification of the natural numbers.
Partition regularity: if $S$ is piecewise syndetic and $S=C_{1}\cup C_{2}\cup \dots \cup C_{n}$ , then for some $i\leq n$ , $C_{i}$ contains a piecewise syndetic set. (Brown, 1968)
If A and B are subsets of $\mathbb {N}$ with positive upper Banach density, then $A+B=\{a+b:a\in A,\,b\in B\}$ is piecewise syndetic.^[1]