In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a syndetic set is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s 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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
subset
of
:
where
. Thus syndetic sets have "bounded gaps"; for a syndetic set
, there is an
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
such that
for any
.
See also
*
Ergodic Ramsey theory Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.
History
Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof that a set of positive upper density ...
*
Piecewise syndetic set
*
Thick set In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set T, for every p \in \mathbb, there is some n \in \mathbb such that \ \subset T.
Examples
Trivially \mathbb is a thick set. Other w ...
References
*
*
* {{cite journal
, last1=Bergelson , first1=Vitaly , authorlink1=Vitaly Bergelson
, last2=Hindman , first2=Neil
, title=Partition regular structures contained in large sets are abundant
, journal=
Journal of Combinatorial Theory
The ''Journal of Combinatorial Theory'', Series A and Series B, are mathematical journals specializing in combinatorics and related areas. They are published by Elsevier. ''Series A'' is concerned primarily with structures, designs, and applicat ...
, series=Series A
, volume=93
, issue=1
, date=2001
, pages=18—36
, doi=10.1006/jcta.2000.3061 , doi-access=free
Semigroup theory
Ergodic theory