Abstract
This thesis is a contribution to arithmetic combinatorics. We present the GreenTao method and the GreenTaoZiegler theorem concerning asymptotics for linear configura tions of primes. Then we show extensions of our own to this theorem: first to some family of quadratic configurations, and secondly to configurations with unbounded coefficients. As a result, we are able to provide an asymptotic for configurations of primes inside the set of shifted squarefree numbers.We then leave integers and move to vector spaces over finite fields. In this context, we prove a bidirectional additive smoothing result for sets of pairs P ⊂ Fnp × Fnp . This is a bilinear version of Bogolyubov’s theorem. We then equip these vector spaces with a multiplicative structure, that is, we consider polynomial rings over finite fields. Using the CrootLevPach method, we show that sets of polynomials of degree less than n that contain no nontrivial solution to a given polynomial equation (of some specific type) is exponentially small.
Finally, we seek to apply the GreenTao method on polynomial rings, with the intention of deriving asymptotics for configurations of irreducible polynomials. To this aim, we bound correlations of the M ̈obius function with linear and quadratic phases.
Date of Award  25 Sep 2018 

Original language  English 
Awarding Institution 

Supervisor  Julia Wolf (Supervisor) 