A multidimensional Szemerédi's theorem in the primes

In this thesis, we investigate topics related to the Green-Tao theorem on arithmetic progression in primes in higher dimensions. Our main tool is the pseudorandom measure majorizing primes defined in [51] concentrated on almost primes. In chapter 2, we combine the sieve technique used in constructing pseudorandom measure (in this case, Goldston-Yildirim sum and almost primes) with the circle method of Birch to study the number of almost prime solutions of diophantine systems (with some rank conditions). Our rank condition is similar to the integer case, due to the heuristics that almost primes are pseudorandom. In chapter 3, we investigate the generalization of Green-Tao’s theorem to higher dimensions in the case of corner configuration. We apply the transference principle of Green-Tao (with hyperplane separation technique of Gowers) in this setting. This problem is also related to the densification trick in [16]. In chapter 4, we extend the result of Chapter 3 to obtain the full multi-dimensional analogue of the Green-Tao’s theorem, using hypergraph regularity method by directly proving a version of hypergraph removal lemma in the weighted hypergraphs. The method is to run an energy increment on a parametric weight systems of measures, rather than on a single measure space, to overcome the presence of intermediate weights. Contrary to [110], [68] where the authors investigate the problem using a measure supported on primes and infinite linear form conditions, relying on the Gowers Inverse Norms Conjecture.Science, Faculty ofMathematics, Department ofGraduat