Thick set
Encyclopedia
In mathematics
, a thick set is a set of integer
s that contains arbitrarily long intervals
. That is, given a thick set
, for every 
, there is some 
such that 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a thick set is a set of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s that contains arbitrarily long intervals
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
. That is, given a thick set
, for every , there is some such that .See also
- Syndetic setSyndetic setIn 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:...
- Piecewise syndetic set
- Ergodic Ramsey theoryErgodic Ramsey theoryErgodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, when Hillel...
- Cofinite set
- Cofinal set