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 there is some fixed bound b and S contains arbitrarily long intervals with gaps bounded by b.

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[edit] Properties

  • 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, Ci contains a piecewise syndetic set. (Brown, 1968)
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