On The Existence of Arithmetic Progressions In Subsets of Integers

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Malekian, Reihaneh | 2018

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 51432 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Alishahi, Kasra; Hatami, Omid
Abstract:
Suppose that A is a large subset of N. It is interesting to think about the arithmetic progressions in A.In 1936, Erdos and Turan conjectured that for > 0 and k 2 N, there exists N = N(k; ) that for all subsets A {1; 2; : : : ;N}, if lAl N, A has a nontrivial arithmetic progression of length k. Roth proved the conjecture for k = 3 in 1953. In 1969, Szemeredi proved the case k = 4 and in 1975, he gave a combinatorial proof for the general case. In 1977, using ergodic theory, Furstenberg gave a different proof for the Erdos-Turan conjecture (or Szemeredi Theorem!) and finally Gowers found another proof for the Szemeredi theorem, which was an elegant generalization of the Roth’s proof for k = 3. If P denotes the set of prime numbers, by prime number theorem, we know that the density of P in N is zero. However, in 2004, Green and Tao proved that P has arithmetic progressions of arbitrary lengths.In this thesis, firstly, we review a proof for Szemeredi theorem in the cases k = 3 and k = 4 and then the proof of Green–Tao theorem
Keywords:
Prime Numbers Theorem ; Szemeredi Theorem ; Green-Tao Theorem ; Arithmetic Progression