# 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 $S=C_{1}\cup C_{2}\cup ...\cup C_{n}$ , then for some $i\leq n$ , $C_{i}$  contains a piecewise syndetic set. (Brown, 1968)
• If A and B are subsets of $\mathbb {N}$ , and A and B have positive upper Banach density, then $A+B=\{a+b:a\in A,b\in B\}$  is piecewise syndetic

## Other notions of largeness

There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers: