Piecewise syndetic set

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

Let $mathcal\{P\}\_f(mathbb\{N\})$ denote the set of finite subsets of $mathbb\{N\}$. Then a set $S\; sub\; mathbb\{N\}$ is called "piecewise syndetic" if there exists $G\; in\; mathcal\{P\}\_f(mathbb\{N\})$ such that for every $F\; in\; mathcal\{P\}\_f(mathbb\{N\})$ there exists an $x\; in\; mathbb\{N\}$ such that

:

where $S-n\; =\; \{m\; in\; mathbb\{N\}:\; m+n\; in\; S\; \}$. Informally, "S" is piecewise syndetic if "S" contains arbitrarily long intervals with gaps bounded by some fixed bound "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 $S$ 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 [R. Jin, [http://math.cofc.edu/faculty/jin/research/banach.pdf Nonstandard Methods For Upper Banach Density Problems] , "Journal of Number Theory" 91, (2001), 20-38.]

Other Notions of Largeness

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

* cofinite
* positive upper density
* syndetic
* thick
* member of a nonprincipal ultrafilter
* IP set

See also

*Ergodic Ramsey theory

References

* J. McLeod, " [http://www.mtholyoke.edu/~jmcleod/somenotionsofsize.pdf Some Notions of Size in Partial Semigroups] " "Topology Proceedings" 25 (2000), 317-332
* V. Bergelson, " [http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf 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)
* V. Bergelson, N. Hindman, " [http://members.aol.com/nhfiles2/pdf/large.pdf Partition regular structures contained in large sets are abundant] ", "J. Comb. Theory (Series A)" 93 (2001), 18-36
* T. Brown, " [http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102971066 An interesting combinatorial method in the theory of locally finite semigroups] ", "Pacific J. Math." 36, no. 2 (1971), 285–289.