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 well-known sets that are thick include non-primes and non-squares. Thick sets can also be sparse, for example:

$\bigcup_ \.$

Generalisations

The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup $(S, \cdot)$ and $A \subseteq S$, $A$ is said to be ''thick'' if for any finite subset $F \subseteq S$, there exists $x \in S$ such that

$F \cdot x = \ \subseteq A.$

It can be verified that when the semigroup under consideration is the natural numbers $\mathbb{N}$ with the addition operation $+$, this definition is equivalent to the one given above.

See also

* Cofinal (mathematics)
* Cofiniteness
* Ergodic Ramsey theory
* Piecewise syndetic set
* Syndetic set

References

* J. McLeod,
Some Notions of Size in Partial Semigroups
, ''Topology Proceedings'', Vol. 25 (Summer 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", ''Journal of Combinatorial Theory, Series A'' 93 (2001), pp. 18-36
* N. Hindman, D. Strauss. ''Algebra in the Stone-Čech Compactification''. p104, Def. 4.45.

Semigroup theory
Ergodic theory