Tree alignment: Difference between revisions
Jump to navigation
Jump to search
en>Rjwilmsi m Journal cites:, added 1 PMIDs, using AWB (7584) |
en>Amkilpatrick |
||
| Line 1: | Line 1: | ||
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]] | |||
Revision as of 22:17, 18 January 2014
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 is called syndetic if for some finite subset F of
where . Thus syndetic sets have "bounded gaps"; for a syndetic set , there is an integer such that for any .
![{\displaystyle [a,a+1,a+2,...,a+p]\bigcap S\neq \emptyset }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f40fe6ece101c6531276a83b3086a9b6bc4b6583)
See also
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