Pseudorandom majorants over number fields with applications in arithmetic geometry

丢番图方程是以古希腊数学家亚历山大·丢番图的名字命名的，它是一个多项式方程，其中所有的系数都是整数(整数)或有理数(分数)。给出一个丢番图方程，最基本的问题是它是否有解，即满足这个方程的整数或有理数的集合。尽管几个世纪以来开发了大量的数学机器来解决这些问题，但要确定给定的丢番图方程是否有解可能是极其困难的。费马大定理就是一个著名的例子。尽管它的陈述相对简单，即对于任何大于2的整数n，两个正n次方的和不可能是n次方，但350多年来数学家们一直没有找到一个证明。它产生了许多新的发展，最终由安德鲁·威尔斯在20世纪末完成。方程不仅定义了数论，还定义了几何对象。20世纪发展起来的一种特别成功的方法，试图通过相应的几何对象来研究丢番图方程的解。使用这些几何技术对丢番图方程进行现代研究的方法被称为算术(或丢番图)几何。数论的另一个分支被称为加性组合学，英国数学家在这一分支中起着世界领先的作用。这门学科的目的之一是通过将整数分解成结构化和随机的部分来理解它们的子集，主要挑战来自这样一个事实，即这通常不是一个干净的二分法，而是一个完整的谱。最近，通过将加法组合学的某些结果和技术应用于算术几何中的问题，这两个领域之间建立了非常富有成效的联系，从而显著扩展了我们对丢番图方程的了解。这个项目的中心目标是通过使这些技术在更广泛的环境中可用来增强这些技术的影响，这在算术几何中是很自然的。

A Diophantine equation, named after the ancient Hellenistic mathematician Diophantus of Alexandria, is a polynomial equation in which all the coefficients are integers (whole numbers) or rational numbers (fractions). The most fundamental question, given a Diophantine equation, is whether it has a solution, that is a collection of integers or rational numbers which satisfy this equation. To decide whether a given Diophantine equation has a solution can be extremely hard, in spite of extensive mathematical machinery that was developed over centuries to attack these questions. A famous example is Fermat's Last Theorem. Despite the relative simplicity of its statement that for any integer n greater than two, the sum of two positive nth powers can not be an nth power, a proof has eluded the efforts of mathematicians for more than 350 years. It has spawned numerous new developments and was finally completed by Andrew Wiles at the end of the 20th century.Equations define not just number theoretic, but also geometric objects. A particularly successful approach, developed in the 20th century, tries to investigate solutions to Diophantine equations via the corresponding geometric objects. The modern study of Diophantine equations using these geometric techniques is called arithmetic (or Diophantine) geometry.Another branch of number theory, in which UK mathematicians play a world leading role, is called additive combinatorics. One of the aims of this discipline is to understand subsets of the integers by decomposing them into structured and random looking parts, with the main challenge arising from the fact that this is usually not a clean dichotomy, but rather a full spectrum.Extremely fruitful connections between these two fields were initiated very recently by applying certain results and techniques from additive combinatorics to questions in arithmetic geometry, thus expanding our knowledge of Diophantine equations significantly. The central aim of this project is to enhance the impact of these techniques by making them available in a much wider context that is natural in arithmetic geometry.