Equidistribution of exponential sums indexed by a subgroup of fixed cardinality

TL;DR

This paper investigates the distribution of exponential sums associated with fixed-order subgroups modulo prime powers, establishing equidistribution results in complex regions described by Laurent polynomials and geometric shapes.

Contribution

It provides new equidistribution results for exponential sums indexed by subgroups of fixed order, with explicit geometric descriptions of the distribution regions.

Findings

Regions of equidistribution are described by Laurent polynomials.

Distribution regions include hypocycloids and Minkowski sums.

Results hold for prime power moduli with fixed subgroup order.

Abstract

We consider families of exponential sums indexed by a subgroup of invertible classes modulo some prime power $q$. For fixed $d$, we restrict to moduli $q$ so that there is a unique subgroup of invertible classes modulo $q$ of order $d$. We study distribution properties of these families of sums as $q$ grows and we establish equidistribution results in some regions of the complex plane which are described as the image of a multi-dimensional torus via an explicit Laurent polynomial. In some cases, the region of equidistribution can be interpreted as the one delimited by a hypocycloid, or as a Minkowski sum of such regions.