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

==Definition==
A set <math>S \sub \mathbb{N}</math> is called syndetic if for some finite subset ''F'' of <math>\mathbb{N}</math>

:<math>\bigcup_{n \in F} (S-n) = \mathbb{N}</math>

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

==See also==
* [[Ergodic Ramsey theory]]
* [[Piecewise syndetic set]] 
* [[Thick set]]

== References ==
* J. McLeod, "[http://www.mtholyoke.edu/%7Ejmcleod/somenotionsofsize.pdf Some Notions of Size in Partial Semigroups]", ''Topology Proceedings, Vol. '''25''' (2000), pp. 317-332
* [[Vitaly Bergelson]], "[http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf 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, "[http://members.aol.com/nhfiles2/pdf/large.pdf Partition regular structures contained in large sets are abundant]", ''J. Comb. Theory (Series A)'' '''93''' (2001), pp. 18-36

[[Category:Semigroup theory]]
[[Category:Ergodic theory]]