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dc.contributor.authorNishizawa, Yui 17:22:39 (GMT) 17:22:39 (GMT)
dc.description.abstractFor any finite set B and a subset A⊆B, we define the density of A in B to be the value α=|A|/|B|. Roth's famous theorem, proven in 1953, states that there is a constant C>0, such that if A⊆{1,...,N} for a positive integer N and A has density α in {1,...,N} with α>C/loglog N, then A contains a non-trivial arithmetic progression of length three (3AP). The proof of this relies on the following dichotomy: either 1) A looks like a random set and the number of 3APs in A is close to the probabilistic expected value, or 2) A is more structured and consequently, there is a progression P of about length α√N on which A∩P has α(1+cα) for some c>0. If 1) occurs, then we are done. If 2) occurs, then we identify P with {1,...,|P|} and repeat the above argument, whereby the density increases at each iteration of the dichotomy. Due to the density increase in case 2), an argument of this type is called a density increment argument. The density increment is obtained by studying the Fourier transforms of the characterstic function of A and extracting a structure out of A. Improving the lower bound for α is still an active area of research and all improvements so far employ a density increment. Two of the most recent results are α>C(loglog N/log N)^{1/2} by Bourgain in 1999 and α>C(loglog N)^5/log N by Sanders in 2010. This thesis is a survey of progresses in Roth's theorem, with a focus on these last two results. Attention was given to unifying the language in which the results are discussed and simplifying the presentation.en
dc.publisherUniversity of Waterlooen
dc.subjectNumber theoryen
dc.subjectAdditive combinatoricsen
dc.titleA survey of Roth's Theorem on progressions of length threeen
dc.typeMaster Thesisen
dc.subject.programPure Mathematicsen Mathematicsen
uws-etd.degreeMaster of Mathematicsen

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