Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres

TL;DR

This paper derives asymptotic formulas for lattice points on certain arithmetic spheres defined by regularly varying functions, and uses these results to analyze ergodic averages and equidistribution properties in a measure-theoretic setting.

Contribution

It introduces new asymptotic formulas for lattice points on generalized spheres and applies them to ergodic theorems and equidistribution problems for these sets.

Findings

Asymptotic formulas for lattice points in arithmetic spheres with regularly varying functions.

Convergence results for ergodic averages over these spheres.

Results on equidistribution of lattice points on the defined spheres.

Abstract

We establish an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{h_1, h_2, h_3}(\lambda): =\{x\in\mathbb Z_+^3:\lfloor h_1(x_1)\rfloor+\lfloor h_2(x_2)\rfloor+\lfloor h_3(x_3)\rfloor=\lambda\} \quad \text{with}\quad \lambda\in\mathbb Z_+; \] where functions $h_1, h_2, h_3$ are constant multiples of regularly varying functions of the form $h(x):=x^c\ell_h(x)$, where the exponent $c>1$ (but close to $1$) and a function $\ell_h(x)$ is taken from a certain wide class of slowly varying functions. Taking $h_1(x)=h_2(x)=h_3(x)=x^c$ we will also derive an asymptotic formula for the number of lattice points in the sets \[ \mathbf S_{c}^3(\lambda) := \{x \in \mathbb Z^3 : \lfloor |x_1|^c \rfloor + \lfloor |x_2|^c \rfloor + \lfloor |x_3|^c \rfloor= \lambda \} \quad \text{with}\quad \lambda\in\mathbb Z_+; \] which can be thought of as a perturbation of the classical Waring problem in three variables. We will use the latter asymptotic formula to study, the main results of this paper, norm and pointwise convergence of the ergodic averages \[ \frac{1}{\#\mathbf S_{c}^3(\lambda)}\sum_{n\in \mathbf S_{c}^3(\lambda)}f(T_1^{n_1}T_2^{n_2}T_3^{n_3}x) \quad \text{as}\quad \lambda\to\infty; \] where $T_1, T_2, T_3:X\to X$ are commuting invertible and measure-preserving transformations of a $\sigma$-finite measure space $(X, \nu)$ for any function $f\in L^p(X)$ with $p>\frac{11-4c}{11-7c}$. Finally, we will study the equidistribution problem corresponding to the spheres $\mathbf S_{c}^3(\lambda)$.