Fourier analytic techniques in finite fields

傅里叶变换将一个函数分解为简单的波状函数的和，在数学和科学领域有许多应用。有一个类似的“离散傅立叶变换”（定义在有限对象上，如有限域上的向量空间），它在各种计数问题上有强大的应用。例如，一个有限向量空间的子集要多大才能确保它包含所有可能三角形的正比例（包括旋转和平移）？PI最近在经典傅立叶分析中开发了一个强大的工具，被称为“傅立叶谱”，他已经成功地用它来解决分形几何中的问题（这里的对象是无限的，“计数问题”呈现出相当不同的风格）。这个项目的目的是在这两个世界之间架起一座桥梁，形成PI最近工作的离散类比，并将它们应用于（有限）计数问题。

The Fourier transform decomposes a function as a sum of simple wave-like functions and has numerous applications across mathematics and science. There is an analogous 'discrete Fourier transform' (defined on a finite object such as a vector space over a finite field) which has powerful applications to various counting problems. For example, how large does a subset of a finite vector space have to be to ensure it contains a positive proportion of all possible triangles (up to rotation and translation)? The PI has recently developed a powerful tool in classical Fourier analysis, known as the 'Fourier spectrum', which he has used successfully to tackle problems in fractal geometry (here the objects are infinite and the 'counting problems' take on a rather different flavour). The aim of this project is to bridge these two worlds and formulate discrete analogues of the PI's recent work and apply them to (finite) counting problems.