Linear, bilinear and polynomial structures in function fields and the primes

Abstract

This thesis is a contribution to arithmetic combinatorics. We present the Green-Tao method and the Green-Tao-Ziegler 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 Croot-Lev-Pach 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 Green-Tao 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 Sept 2018
Original language	English
Awarding Institution	University of Bristol
Supervisor	Julia Wolf (Supervisor)