Piecewise syndetic set explained

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

A set

S \sub N is called piecewise syndetic if there exists a finite subset G of N such that for every finite subset F of N there exists an x \in N such thatx + F \subset c up _{n} (S - n)

where

S - n = \{m \in N : m + n \in S \} . Equivalently, S is piecewise syndetic if there is a constant b such that there are arbitrarily long intervals of N where the gaps in S are bounded by 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 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

N with 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

See also

• Ergodic Ramsey theory

References

• McLeod . Jillian. Some Notions of Size in Partial Semigroups. Topology Proceedings. 25. Summer 2000. 2000. 317—332.
• Book: Bergelson . Vitaly . Vitaly Bergelson. Minimal Idempotents and Ergodic Ramsey Theory. Topics in Dynamics and Ergodic Theory. 8—39. London Mathematical Society Lecture Note Series. 310. Cambridge University Press, Cambridge. 2003. 10.1017/CBO9780511546716.004. http://www.math.ohio-state.edu/~vitaly/vbkatsiveli20march03.pdf.
• Bergelson . Vitaly . Vitaly Bergelson . Hindman . Neil . Neil Hindman . Partition regular structures contained in large sets are abundant . . Series A . 93 . 1 . 2001 . 18—36 . 10.1006/jcta.2000.3061 . free.
• Brown . Thomas Craig. An interesting combinatorial method in the theory of locally finite semigroups. Pacific Journal of Mathematics. 36. 2. 1971. 285—289. 10.2140/pjm.1971.36.285 . free.

Notes and References

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