Convolution and Equidistribution explores an important aspect of
number theory--the theory of exponential sums over finite fields and
their Mellin transforms--from a new, categorical point of view. The book
presents fundamentally important results and a plethora of examples,
opening up new directions in the subject.
The finite-field Mellin transform (of a function on the multiplicative
group of a finite field) is defined by summing that function against
variable multiplicative characters. The basic question considered in the
book is how the values of the Mellin transform are distributed (in a
probabilistic sense), in cases where the input function is suitably
algebro-geometric. This question is answered by the book's main theorem,
using a mixture of geometric, categorical, and group-theoretic methods.
By providing a new framework for studying Mellin transforms over finite
fields, this book opens up a new way for researchers to further explore
the subject.