Exploring the interface of analytic number theory, additive combinatorics and Diophantine geometry

This research project is part of a long-term project with the aim of resolving a conjecture of Colliot-Thélène which belongs to the area of Diophantine geometry.

Our main tools will however come from the areas of additive combinatorics and analytic number theory.A completely new connection between methods from additive combinatorics and algebraic geometry was opened up in previous joint work of the project proposer with Tim Browning and Alexei Skorobogatov and has lead to the first significant progress on a related conjecture of Colliot-Thélène and Sansuc in more than a decade.New work of the proposer on multiplicative functions and its arithmetic application in joint work of the proposer and Daniel Loughran, proving for the very first time correct-order lower bounds about certain counting functions attached to fairly general fibrations, has highlighted the importance that general multiplicative functions play in the study of rational points.Based on our new insights from the past few years and new developments within analytic number theory and additive combinatorics, our aim here is to push the methods far enough to obtain the first results that go beyond the Green-Tao-based approach from our previous work, which will be a significant step forward.

The impact that our previous work on the above-mentioned conjectures had and the activity and new work to which it lead within the community show clearly the importance of this research.