Ergotheoretical proof of Szemeredi’s theorem

Postgraduate Thesis uoadl:2941877 59 Read counter

Unit:
Κατεύθυνση Θεωρητικά Μαθηματικά
Library of the School of Science
Deposit date:
2021-04-05
Year:
2021
Author:
Iakovidis Isidoros
Supervisors info:
Γατζούρας Δημήτριος, Καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ
Γιαννόπουλος Απόστολος, Καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ
Δοδός Παντελής, Αναπληρωτής καθηγητής, Μαθηματικό τμήμα, ΕΚΠΑ
Original Title:
Ergotheoretical proof of Szemeredi’s theorem
Languages:
English
Translated title:
Ergotheoretical proof of Szemeredi’s theorem
Summary:
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.
Main subject category:
Science
Keywords:
Furstenberg 's theorem , Szemeredi's theorem, ergodic theroy, Sarkozy theorem,recurrence
Index:
No
Number of index pages:
0
Contains images:
No
Number of references:
16
Number of pages:
109
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