Existence of Arithmetic Progressions in Subsets of Natural Numbers

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Zareh Bidaki, Mojtaba | 2021

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 54603 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Rastegar, Arash; Hatami Varzaneh, Omid
Abstract:
Szemeredi's theorem is one of the significant theorems in additive combinatorics which was started by Van Der Waerden's theorem in 1927. Erdos and Turan conjectured generalized versions of Van Der Waerden's theorem in several ways included Szemeredi's theorem. In 1975 Szemeredi proved the conjecture using complicated combinatorial methods. In 1977 H. Furstenberg proved Szemeredi's theorem via the Ergodic theory approach which led to prove polynomial Szemeredi's theorem and multi-dimensional Szemeredi's theorem. The Ergodic approach is the only known approach so far to these generalizations of this theorem which is named Ergodic Ramsey theory and led to some other problems in Ergodic theory like the limit behavior of non-conventional ergodic sums. In this thesis, we explore Furstenberg's proof and other generalizations included non-conventional ergodic sums.
Keywords:
Arithmetic Progression ; Ergodic Theory ; Szemeredi Theorem ; Roth's Theorem