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

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