Piecewise syndetic set

In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers.

A set is called piecewise syndetic if there exists a finite subset G of such that for every finite subset F of there exists an such that

where . Equivalently, S is piecewise syndetic if there are arbitrarily long intervals of where the gaps in S are bounded by some constant b.

Properties

• A set is piecewise syndetic if and only if it is the intersection of a syndetic set and a thick set.
• If S is piecewise syndetic then S contains arbitrarily long arithmetic progressions.
• A set S is piecewise syndetic if and only if there exists some ultrafilter U which contains S and U is in the smallest two-sided ideal of , the Stone–Čech compactification of the natural numbers.
• Partition regularity: if is piecewise syndetic and , then for some , contains a piecewise syndetic set. (Brown, 1968)
• If A and B are subsets of , and A and B have positive upper Banach density, then is piecewise syndetic^[1]

Other Notions of Largeness

There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers:

• Cofiniteness
• IP set
• member of a nonprincipal ultrafilter
• positive upper density
• syndetic set
• thick set

See also

• Ergodic Ramsey theory

Notes

References

• J. McLeod, "Some Notions of Size in Partial Semigroups" Topology Proceedings 25 (2000), 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), 18-36
• T. Brown, "An interesting combinatorial method in the theory of locally finite semigroups", Pacific J. Math. 36, no. 2 (1971), 285–289.