Piecewise syndetic set
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.
- 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
Other Notions of Largeness
There are many alternative definitions of largeness that also usefully distinguish subsets of natural numbers:
- IP set
- member of a nonprincipal ultrafilter
- positive upper density
- syndetic set
- thick set
- R. Jin, Nonstandard Methods For Upper Banach Density Problems, Journal of Number Theory 91, (2001), 20-38.
- 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.