mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, piecewise syndeticity is a notion of largeness of

subset In mathematics, 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 are ...

s of the

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s. A set

S \sub \mathbb

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

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * 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 ...

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 Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...

\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 i ...

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 w ...

. * 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 between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

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

ultrafilter In the 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 filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...

''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 map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Stone ...

of the natural numbers. *

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

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 it is countable, then one says the set is cocoun ...

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 In number theory, natural density (also referred to as asymptotic density or arithmetic density) is one method to measure how "large" a subset of the set of natural numbers is. It relies chiefly on the probability of encountering members of the de ...

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 organisati ...

, 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