Thick set explained

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 N , there is some n \in N such that \{n, n +1 , n +2 , ... , n + p \} \subset T .

Examples

Trivially

N 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:

$$\backslash bigcup\_\; \backslash .$$

Generalisations

The notion of a thick set can also be defined more generally for a semigroup, as follows. Given a semigroup

(S, ⋅ ) 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

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

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

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.