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

Examples

Trivially $\mathbb {N}$ 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:

$\bigcup _{n\in \mathbb {N} }\{x:x=10^{n}+m:0\leq m\leq n\}$ .

References

J. McLeod, "Some Notions of Size in Partial Semigroups", Topology Proceedings, Vol. 25 (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", J. Comb. Theory (Series A) 93 (2001), pp. 18-36

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Thick set

Examples

See also

References