mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, piecewise syndeticity is a notion of largeness of

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

s of the

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

s. A set

S \sub \mathbb

is called ''piecewise syndetic'' if there exists a

finite Finite may refer to: * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Gr ...

subset ''G'' of

\mathbb

such that for every finite subset ''F'' of

\mathbb

there exists an

x \in \mathbb

such that :

x+F \subset \bigcup_ (S-n)

where

S-n = \

. Equivalently, ''S'' is piecewise syndetic if there is a constant ''b'' such that there are arbitrarily long intervals of

\mathbb

where the gaps in ''S'' are bounded by ''b''.

Properties

* A set is piecewise syndetic if and only if it is the

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

of a

syndetic set In mathematics, a syndetic set is a subset of the natural numbers having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded. Definition A set S \sub \mathbb is called syndetic if for some finit ...

and a

thick set In mathematics, a thick set is a set of integers that contains arbitrarily long intervals. That is, given a thick set T, for every p \in \mathbb, there is some n \in \mathbb such that \ \subset T. Examples Trivially \mathbb is a thick set. Other ...

. * If ''S'' is piecewise syndetic then ''S'' contains arbitrarily long

arithmetic progression An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that ...

s. * A set ''S'' is piecewise syndetic if and only if there exists some

ultrafilter In the Mathematics, mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a Maximal element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...

''U'' which contains ''S'' and ''U'' is in the smallest two-sided ideal of

\beta \mathbb

, the

Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a Universal property, universal map from a topological space ''X'' to a Compact space, compact Ha ...

of the natural numbers. *

Partition regular In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a set system, collection of sets. Given a set X, a collection of subsets \mathbb \subset \mathcal(X) is called ''partition regular'' if every set ''A'' ...

ity: if

S

is piecewise syndetic and

S = C_1 \cup C_2 \cup \dots \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

with positive upper Banach density, then

A+B=\

is piecewise syndetic.R. Jin
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: *

Cofiniteness In mathematics, a cofinite subset of a set X is a subset A whose complement in X is a finite set. In other words, A contains all but finitely many elements of X. If the complement is not finite, but is countable, then one says the set is cocounta ...

IP set In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set ''D'' of natural numbers are all those numbers that can be obtained by adding up the elements of some finite nonem ...

* member of a nonprincipal

* positive upper density *

Notes

References

* * * * {{cite journal , last1=Brown , first1=Thomas Craig , url=http://projecteuclid.org/euclid.pjm/1102971066 , title=An interesting combinatorial method in the theory of locally finite semigroups , journal=

Pacific Journal of Mathematics The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisa ...

, volume=36 , issue=2 , date=1971 , pages=285–289 , doi=10.2140/pjm.1971.36.285 , doi-access=free Semigroup theory Ergodic theory Ramsey theory Combinatorics

Properties

Other notions of largeness

See also

Notes

References