Distribution of gaps between eigenangles of Hecke operators

Abstract

In 1931, Van der Corput showed that if for each positive integer $s$, the sequence $\{x_{n+s}-x_n\}$ is uniformly distributed (mod 1), then the sequence $x_n$ is uniformly distributed (mod 1). The converse of above result is surprisingly not true. The distribution of consecutive gaps of an equidistributed sequence has been studied widely in the literature. In this paper, we have studied the distribution of gaps between one or more equidistributed sequences. Under certain conditions, we could study the distribution effectively. As applications, we study the equidistribution of gaps between eigenangles of Hecke operators acting on space of cusp forms of weight $k$ and level $N$, primitive Maass forms. We also have studied the distribution of gaps between corresponding angles of Satake parameters of $GL_2$ with prescribed local representations.