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An ergodic theoretic approach to Szemerédi's theorem
Dissertation (MSc)--University of Pretoria, 2018.
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| Format: | Thesis |
| Language: | English |
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University of Pretoria
2019
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| _version_ | 1867613642189438976 |
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| access_status_str | Open Access |
| author2 | Van der Walt, Jan Harm |
| author_browse | Van der Walt, Jan Harm |
| author_facet | Van der Walt, Jan Harm |
| collection | Thesis |
| dc_rights_str_mv | © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. |
| description | Dissertation (MSc)--University of Pretoria, 2018. |
| format | Thesis |
| id | oai:repository.up.ac.za:2263/70516 |
| institution | University of Pretoria (South Africa) |
| language | English |
| last_indexed | 2026-06-10T12:39:23.523Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UPSpace — University of Pretoria Institutional Repository |
| publishDate | 2019 |
| publishDateRange | 2019 |
| publishDateSort | 2019 |
| publisher | University of Pretoria |
| publisherStr | University of Pretoria |
| record_format | dspace |
| source_str | UPSpace — University of Pretoria Institutional Repository |
| spelling | oai:repository.up.ac.za:2263/70516 An ergodic theoretic approach to Szemerédi's theorem Van der Walt, Jan Harm u13044339@tuks.co.za Messerschmidt, Miek Van Amstel, Sarel Jakobus van der Walt UCTD Dissertation (MSc)--University of Pretoria, 2018. In this dissertation, Szemer edi's Theorem is proven using ergodic theoretic techniques via the Furstenberg Multiple Recurrence Theorem. Brief historical remarks, along with a non-technical layout of the ideas behind the proof of the Furstenberg Multiple Recurrence Theorem, are given in Chapter 1. After introducing some notation, preliminary de nitions and propositions in Chapter 2, the equivalence of the Furstenberg Multiple Recurrence Theorem and Szemer edi's Theorem is laid out in detail in Chapter 3. The rest of this work is devoted to providing a proof of the Furstenberg Multiple Recurrence Theorem. Two important classes of invertible measure preserving systems, weak mixing and compact systems, are introduced in Chapters 4 and 5 respectively, where it is shown that these classes of measure preserving systems satisfy the Furstenberg Multiple Recurrence Theorem. (We shall say these systems have the Furstenberg property). In Chapter 6, a dichotomy result is proven that characterizes all invertible measure preserving systems in terms of weak mixing and compact systems. After introducing more preliminary de nitions and propositions in Chapter 7, a short proof of Roth's Theorem, the rst non-trivial special case of Szemer edi's Theorem, is given in Chapter 8. In Chapter 9, a generalization of weak mixing systems, known as weak mixing extensions, is introduced. It is shown that if a measure preserving Y has the Furstenberg property and X is a weak mixing extension of Y, the Furstenberg property passes through the extension to the extended system X. The analogous generalization of compact systems - compact extensions - is introduced in Chapter 10 and it is shown that the Furstenberg property passes through compact extensions. Similar to what was done in Chapter 6, a dichotomy result is proven in Chapter 11 that characterizes extensions of invertible measure preserving systems in terms of weak mixing and compact extensions. All of the necessary tools developed in previous chapters are put to use in Chapter 12 where the Furstenberg Multiple Recurrence Theorem is proven - thus establishing Szemer edi's Theorem. Mathematics and Applied Mathematics MSc Unrestricted 2019-07-08T09:46:44Z 2019-07-08T09:46:44Z 2019/04/09 2018 Dissertation Van Amstel, SJVDW 2018, An ergodic theoretic approach to Szemerédi's theorem, MSc Dissertation, University of Pretoria, Pretoria, viewed yymmdd <http://hdl.handle.net/2263/70516> A2019 http://hdl.handle.net/2263/70516 en © 2019 University of Pretoria. All rights reserved. The copyright in this work vests in the University of Pretoria. No part of this work may be reproduced or transmitted in any form or by any means, without the prior written permission of the University of Pretoria. application/pdf University of Pretoria |
| spellingShingle | UCTD An ergodic theoretic approach to Szemerédi's theorem |
| title | An ergodic theoretic approach to Szemerédi's theorem |
| title_full | An ergodic theoretic approach to Szemerédi's theorem |
| title_fullStr | An ergodic theoretic approach to Szemerédi's theorem |
| title_full_unstemmed | An ergodic theoretic approach to Szemerédi's theorem |
| title_short | An ergodic theoretic approach to Szemerédi's theorem |
| title_sort | ergodic theoretic approach to szemeredi s theorem |
| topic | UCTD |
| url | http://hdl.handle.net/2263/70516 |