Piecewise syndetic set

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

Let $\mathcal{P}_f(\mathbb{N})$ denote the set of finite subsets of $\mathbb{N}$ . Then a set $S \sub \mathbb{N}$ is called piecewise syndetic if there exists $G \in \mathcal{P}_f(\mathbb{N})$ such that for every $F \in \mathcal{P}_f(\mathbb{N})$ there exists an $x \in \mathbb{N}$ such that

$x+F \subset \bigcup_{n \in G} (S-n)$

where $S-n = \{m \in \mathbb{N}: m+n \in S \}$ . Informally, S is piecewise syndetic if S contains arbitrarily long intervals with gaps bounded by some fixed bound b.

[edit] 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^[1]