Syndetic set

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In mathematics, a syndetic set is a subset of the natural numbers, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.

[edit] Definition

Let \mathcal{P}_f(\mathbb{N}) denote the set of finite subsets of \mathbb{N}. Then a set S \sub \mathbb{N} is called syndetic if for some F \in \mathcal{P}_f(\mathbb{N})

\bigcup_{n \in F} (S-n) = \mathbb{N}

where S-n = \{m \in \mathbb{N} : m+n \in S \}. Thus syndetic sets have "bounded gaps"; for a syndetic set S, there is an integer p = p(S) such that [a, a+1, a+2, ... , a+p] \bigcap S \neq \emptyset for any a \in \mathbb{N}.

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