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

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

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

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