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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.

Definition

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

${\displaystyle \bigcup _{n\in F}(S-n)=\mathbb {N} }$

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

See also

• Ergodic Ramsey theory
• Piecewise syndetic set
• Thick set

References

• McLeod, Jillian (2000). "Some Notions of Size in Partial Semigroups" (PDF). Topology Proceedings. 25 (Summer 2000): 317–332.
• Bergelson, Vitaly (2003). "Minimal Idempotents and Ergodic Ramsey Theory" (PDF). Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series. Vol. 310. Cambridge University Press, Cambridge. pp. 8–39. doi:10.1017/CBO9780511546716.004. ISBN 978-0-521-53365-2.
• Bergelson, Vitaly; Hindman, Neil (2001). "Partition regular structures contained in large sets are abundant". Journal of Combinatorial Theory. Series A. 93 (1): 18–36. doi:10.1006/jcta.2000.3061.

• Semigroup theory
• Ergodic theory