Asymptotic evaluations of generalized Bessel function of order zero related to the p-circle lattice point problem

TL;DR

This work extends the harmonic-analytic approach to lattice-point problems from circles ($p=2$) to general p-circles (Lamé curves) by introducing and analyzing generalized Bessel functions $J_{\omega}^{[p]}$. The authors derive oscillatory-integral representations for $J_{0}^{[p]}$ and establish two main uniform asymptotic regimes: (i) uniform decay on compact sets in each quadrant for $0<p\le1$ or $p=2$ with rate $|\eta|_{p}^{-1/2}$, and (ii) uniform decay on all of $\mathbb{R}^{2}$ when $\frac{2}{p}$ is a natural number, with rate $|\eta|_{p}^{-p/2}$ along axis directions. The proofs combine stationary-phase analysis, endpoint considerations, and axis-neighborhood perturbations, and they lay groundwork for extending the method to general $p$ via oscillatory representations of higher-order $J_{\omega}^{[p]}$, ultimately aiming at broader lattice-point error bounds for Lamé curves. The paper also discusses future steps, including identifying integrability thresholds for related kernels and formulating Laurent-based representations to systematize the $J_{\omega}^{[p]}$-driven expansion of the lattice-point error.

Abstract

Let $p$ and $r$ be positive real numbers. Then, we consider the lattice point problem of the closed curve $p$-circle $\{x\in\mathbb{R}^{2}|\ |x_{1}|^{p}+|x_{2}|^{p}=r^{p}\}$ which is a generalization of the circle ($p=2$). Following the harmonic analytic approach of S. Kuratsubo and E. Nakai for the case of a circle, we need to investigate properties of appropriately generalized Bessel functions for $p$ in order to tackle the problem. Thus, in this paper, we derive asymptotic evaluations of the generalized Bessel function of order zero, such as uniformly asymptotic estimates on compact sets on quadrants of $\mathbb{R}^{2}$ for the cases $0<p\leq1$ or $p=2$, and, as stronger results, uniformly asymptotic estimates on $\mathbb{R}^{2}$ for the cases $p$ such that $\frac{2}{p}$ are the natural numbers.

Figures (1)

• Figure 1: Examples of the $p$-circle and the approximation by unit squares.

Theorems & Definitions (21)

• Theorem 1.1: Kratzel, Theorem 3.17 A
• Proposition 1.2: Kuratsubo-2022; Lemma 5.1(Special cases)
• Theorem 1.3: K1, Theorem 1.3
• Theorem 1.4: Uniformly asymptotic estimates on compact sets on quadrants
• Theorem 1.5: Uniformly asymptotic estimates on $\mathbb{R}^{2}$
• Remark 1.6
• Proposition 2.1
• proof
• Remark 2.2
• Lemma 2.3