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 ${\displaystyle S\subset \mathbb {N} }$ is called syndetic if for some finite subset 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 ${\displaystyle [a,a+1,a+2,...,a+p]\bigcap S\neq \emptyset }$ for any ${\displaystyle a\in \mathbb {N} }$.

See also

• Ergodic Ramsey theory
• Piecewise syndetic set
• Thick set

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