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

{{cite journal

| last1=McLeod | first1=Jillian

| title=Some Notions of Size in Partial Semigroups

| journal=Topology Proceedings

| volume=25

| issue=Summer 2000

| date=2000

| pages=317–332

| url=http://topology.nipissingu.ca/tp/reprints/v25/tp25217.pdf}}

{{cite book

| last1=Bergelson | first1=Vitaly | authorlink1=Vitaly Bergelson

| chapter=Minimal Idempotents and Ergodic Ramsey Theory

| title=Topics in Dynamics and Ergodic Theory

| pages=8–39

| series=London Mathematical Society Lecture Note Series

| volume=310

| publisher=Cambridge University Press, Cambridge

| date=2003

| doi=10.1017/CBO9780511546716.004

| isbn=978-0-521-53365-2 | chapter-url=http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf}}

{{cite journal |last1=Bergelson |first1=Vitaly |authorlink1=Vitaly Bergelson |last2=Hindman |first2=Neil |author-link2=Neil Hindman |title=Partition regular structures contained in large sets are abundant |journal=Journal of Combinatorial Theory |series=Series A |volume=93 |issue=1 |date=2001 |pages=18–36 |doi=10.1006/jcta.2000.3061 |doi-access=free}}

Category:Semigroup theory

Category:Ergodic theory

Syndetic set

Definition

See also

References