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

Abstract

We establish an asymptotic formula for the number of lattice points in the sets Sh1,h2,h3(λ):={x∈Z+3:⌊h1(x1)⌋+⌊h2(x2)⌋+⌊h3(x3)⌋=λ} with λ∈Z+; where functions h1, h2, h3 are constant multiples of regularly varying functions of the form h(x) : = xcℓh(x) , where the exponent c> 1 (but close to 1) and a function ℓh(x) is taken from a certain wide class of slowly varying functions. Taking h1(x) = h2(x) = h3(x) = xc we will also derive an asymptotic formula for the number of lattice points in the sets Sc3(λ):={x∈Z3:⌊|x1|c⌋+⌊|x2|c⌋+⌊|x3|c⌋=λ}withλ∈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 1#Sc3(λ)∑n∈Sc3(λ)f(T1n1T2n2T3n3x)asλ→∞;where T1, T2, T3: X→ X are commuting invertible and measure-preserving transformations of a σ-finite measure space (X, ν) for any function f∈ Lp(X) with p>11-4c11-7c. Finally, we will study the equidistribution problem corresponding to the spheres Sc3(λ).