Syndetic set

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.

Definition

A set $S \sub \mathbb{N}$ is called syndetic if for some finite subset F of $\mathbb{N}$

\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 $a \in \mathbb{N}$ .

References

J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (2000), pp. 317-332
Vitaly 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)
Vitaly Bergelson, N. Hindman, "Partition regular structures contained in large sets are abundant", J. Comb. Theory (Series A) 93 (2001), pp. 18-36

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Syndetic set

Definition

See also

References