Distribution of gaps between eigenangles of Hecke operators

TL;DR

This paper investigates the distribution of gaps between eigenangles of Hecke operators and Satake parameters, providing effective analysis under certain conditions and exploring their equidistribution properties.

Contribution

It introduces a new study of gap distributions between eigenangles of Hecke operators and Satake parameters, extending previous work on equidistributed sequences.

Findings

Distribution of gaps studied under certain conditions

Effective analysis of gap distribution achieved

Applications to eigenangles of Hecke operators and Satake parameters

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.