Ergodic Ramsey theory
Encyclopedia
Ergodic Ramsey theory is a branch of mathematics
where problems motivated by additive combinatorics are proven using ergodic theory
.
Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof
that a set of positive upper density contains arbitrarily long arithmetic progression
s, when Hillel Furstenberg gave a new proof of this theorem using ergodic theory. It has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure-preserving dynamical system
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
where problems motivated by additive combinatorics are proven using ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
.
Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof
Szemerédi's theorem
In number theory, Szemerédi's theorem is a result that was formerly the Erdős–Turán conjecture...
that a set of positive upper density contains arbitrarily long arithmetic progression
Arithmetic progression
In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant...
s, when Hillel Furstenberg gave a new proof of this theorem using ergodic theory. It has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure-preserving dynamical system
Measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.-Definition:...
s.
See also
- Arithmetic combinatoricsArithmetic combinatoricsArithmetic combinatorics arose out of the interplay between number theory, combinatorics, ergodic theory and harmonic analysis. It is about combinatorial estimates associated with arithmetic operations...
- Ramsey theoryRamsey theoryRamsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that studies the conditions under which order must appear...
- Syndetic setSyndetic setIn 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:...
- Piecewise syndetic set
- Thick set
- IP setIP setIn 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 nonempty subset of D.The set of all finite sums over D is often...