Piecewise syndetic set

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

A set $$S \sub \mathbb{N}$$ 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.

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


 * Cofiniteness
 * IP set
 * member of a nonprincipal ultrafilter
 * positive upper density
 * syndetic set
 * thick set