Syndetic set

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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}.

[edit] See also

[edit] References

  1. J. McLeod Some Notions of Size in Partial Semigroups Topology Proceedings, Vol. 25 (2000), 317-332
  2. V. Bergelson Minimal Idempotents and Ergodic Ramsey Theory Topics in Dynamics and Ergodic Theory 8-39, London Math. Soc. Lecture Note Series 310, Cambridge Univ. Press, Cambridge, (2003)
  3. V. Bergelson, N. Hindman Partition regular structures contained in large sets are abundant J. Comb. Theory (Series A) 93 (2001), 18-36
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