Visual properties of generalized Kloosterman sums

Abstract: For a positive integer $m$ and a subgroup $\Lambda$ of the unit group $(\mathbb{Z}/m\mathbb{Z})^\times$, the corresponding generalized Kloosterman sum is the function $$K(a,b,m,\Lambda) = \sum_{u \in \Lambda}e(\frac{au + bu^{-1}}{m})$$. Unlike classical Kloosterman sums, which are real valued, generalized Kloosterman sums display a surprising array of visual features when their values are plotted in the complex plane. In a variety of instances, we identify the precise number-theoretic conditions that give rise to particular phenomena.