Lattice point problems, equidistribution and ergodic theorems for certain arithmetic spheres

Abstract: Let $\lambda\in\Z_+$ be a positive integer and define the set $\mathbf S_{2}^3(\lambda)$ of all lattice points on a two-dimensional sphere with radius $\lambda^{1/2}$ by \[ \mathbf S_{2}^3(\lambda) := \{x \in \mathbb Z^3 : x_1^2 +x_2^2 +x_3^2 = \lambda \}. \] The study of the behavior of $\mathbf S_{2}^3(\lambda)$ as $\lambda\to\infty$ is a central problem in number theory, which has gone through a period of considerable change and development in the last three decades.

In the recent work with A. Iosevich, M. Mirek and T.Z. Szarek we consider perturbations of the discrete spheres $\mathbf S_2^3(\lambda)$. In particular, for $c\in (1,2)$ we 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. Then we use the obtained asymptotic formula to study 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 study the equidistribution problem corresponding to the spheres $\mathbf S_{c}^3(\lambda)$.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

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