Fourier analytic techniques in finite fields

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.