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Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2021-04-05

2021

Iakovidis Isidoros

Γατζούρας Δημήτριος, Καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ

Γιαννόπουλος Απόστολος, Καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ

Δοδός Παντελής, Αναπληρωτής καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ

Γιαννόπουλος Απόστολος, Καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ

Δοδός Παντελής, Αναπληρωτής καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ

Ergotheoretical proof of Szemeredi’s theorem

English

Ergotheoretical proof of Szemeredi’s theorem

In this study we give a proof of Szemeredi’s theorem via ergodic theory.

In chapter 1, a brief introduction of the master thesis is given.

In chapter 2, the basic study of ergodic theory is presented where we introduce theorems of re-

currence and ergodic theorems. The notion of mixing measure preserving systems is defined, we

study their properties and finally the ergodic decomposition theorem is proved.

In chapter 3, we provide the necessary probability-measure theory background. In particular the

conditional expectation map is defined, the notion of martigales and the conditional measures on a

measure preserving system.

In chapter 4, we define the notion of a factor map and we prove the equivalence of a sub σ− algebra

in a measure preserving system and a factor map. In addition we define the joinings of a set and in

particular the relatively independent joining.

Finally in order to prove Szemeredi’s theorem, a theorem about arithmetic progressions we need to

translate the problem into a problem of ergodic theory. This is accomplished by Furstenberg’s cor-

respondence principle. Next we prove Szemeredi’s theorem for some specific measure preserving

systems and finally we obtain Szemeredi’s theorem for any measure preserving system.

In chapter 1, a brief introduction of the master thesis is given.

In chapter 2, the basic study of ergodic theory is presented where we introduce theorems of re-

currence and ergodic theorems. The notion of mixing measure preserving systems is defined, we

study their properties and finally the ergodic decomposition theorem is proved.

In chapter 3, we provide the necessary probability-measure theory background. In particular the

conditional expectation map is defined, the notion of martigales and the conditional measures on a

measure preserving system.

In chapter 4, we define the notion of a factor map and we prove the equivalence of a sub σ− algebra

in a measure preserving system and a factor map. In addition we define the joinings of a set and in

particular the relatively independent joining.

Finally in order to prove Szemeredi’s theorem, a theorem about arithmetic progressions we need to

translate the problem into a problem of ergodic theory. This is accomplished by Furstenberg’s cor-

respondence principle. Next we prove Szemeredi’s theorem for some specific measure preserving

systems and finally we obtain Szemeredi’s theorem for any measure preserving system.

Science

Furstenberg 's theorem , Szemeredi's theorem, ergodic theroy, Sarkozy theorem,recurrence

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