# Piecewise syndetic set

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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

## Notes

- ↑ R. Jin, Nonstandard Methods For Upper Banach Density Problems,
*Journal of Number Theory***91**, (2001), 20-38</math>.

## 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.