In mathematics, a thick set is a set of integers that contains arbitrarily long intervals.

Examples

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:

Generalisations

The notion of a thick set can also be defined more generally for a semigroup is said to be thick if for any finite subset such that

It can be verified that when the semigroup under consideration is the natural numbers, 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.