释义 |
- Properties
- Other notions of largeness
- See also
- Notes
- References
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 largenessThere 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 1. ^R. Jin, Nonstandard Methods For Upper Banach Density Problems, Journal of Number Theory 91, (2001), 20-38.
References - J. McLeod, "[https://web.archive.org/web/20061117184558/http://www.mtholyoke.edu/~jmcleod/somenotionsofsize.pdf 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, "[https://projecteuclid.org/euclid.pjm/1102971066 An interesting combinatorial method in the theory of locally finite semigroups]", Pacific J. Math. 36, no. 2 (1971), 285–289.
4 : Semigroup theory|Ergodic theory|Ramsey theory|Combinatorics |