Piecewise syndetic set

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

A set $S\subset {\mathbb {N}}$ is called piecewise syndetic if there exists a finite subset G of $\mathbb {N}$ such that for every finite subset F of $\mathbb {N}$ there exists an $x\in \mathbb {N}$ such that

x+F\subset \bigcup _{n\in G}(S-n)

where $S-n=\{m\in \mathbb {N} :m+n\in S\}$ . Equivalently, S is piecewise syndetic if there are arbitrarily long intervals of $\mathbb {N}$ 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 $\beta \mathbb {N}$ , the Stone–Čech compactification of the natural numbers.
Partition regularity: if $S$ is piecewise syndetic and $S=C_{1}\cup C_{2}\cup ...\cup C_{n}$ , then for some $i\leq n$ , $C_{i}$ contains a piecewise syndetic set. (Brown, 1968)
If A and B are subsets of $\mathbb {N}$ , and A and B have positive upper Banach density, then $A+B=\{a+b:a\in A,b\in B\}$ 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

Notes

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

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.

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[1] R. Jin, Nonstandard Methods For Upper Banach Density Problems, Journal of Number Theory 91, (2001), 20-38.

Piecewise syndetic set

Properties

Other notions of largeness

See also

Notes

References