"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.
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