Almost periodic functions and hyperbolic counting

TL;DR

This paper investigates the statistical properties of certain almost periodic functions and introduces an improved estimate for the hyperbolic circle problem, establishing new results on asymptotic behavior and distributions.

Contribution

It proves the existence of asymptotic moments and tail estimates for a class of almost periodic functions and improves the hyperbolic circle problem estimate on the variance of the remainder.

Findings

Existence of asymptotic moments for specific almost periodic functions

An improved estimate on the tails of the limiting distribution

Enhanced variance estimate for the hyperbolic circle problem

Abstract

In this paper we prove the existence of asymptotic moments, and an estimate on the tails of the limiting distribution, for a specific class of almost periodic functions. Then we introduce the hyperbolic circle problem, proving an estimate on the asymptotic variance of the remainder that improves a result of Chamizo. Applying the results of the first part we prove the existence of limiting distribution and asymptotic moments for three functions that are integrated versions of the remainder, and were considered originally (with due adaptations to our settings) by Wolfe, Phillips and Rudnick, and Hill and Parnovski.