Fourier analytic techniques in finite fields

傅里叶变换将函数分解为简单的波函数之和，并在数学和科学中有许多应用。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），它对各种计数问题有强大的应用。例如，一个有限向量空间的子集必须有多大才能确保它包含所有可能三角形的正比例（直到旋转和平移）？PI最近开发了一个强大的工具，在经典的傅立叶分析，被称为“傅立叶频谱”，他成功地利用它来解决问题的分形几何（在这里的对象是无限的和“计数问题”采取了相当不同的味道）。这个项目的目的是弥合这两个世界，制定离散类似物的PI的最近的工作，并将其应用于（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.