Non elementary methods in combinatorial number theory: Roth's and Sarkozy's theorems

Roth's theorem states that every set A with positive density has an arithmetic progression of length 3, i.e. x, x+r, x+2r are in A. In this work we present two different arguments used to proof Roth's theorem and we translate them to the nonstandard framework. The first argument, called density increment, aims to recursively find arithmetic progressions on which the set A has increased density. The second argument, called energy increment, aims to decompose the set in a "structured" component plus a "random" component. Using the transfer principle we translate the density increment argument to the nonstandard setting where we obtain a slightly easier argument at the cost of losing the estimate found in the standard case. For the energy increment argument, we use the Loeb measure and the conditional expectation in nonstandard context to find a decomposition. In the last chapter we adapt the density increment argument to Sarkozy's theorem (which states that a set of positive density contains two elements whose difference is a perfect square) using an estimate on Weyl sums and a theorem on quadratic recurrence