2021/06/22 by Alex Iosevich, Iosevich, Alex, Bartosz Langowski +5
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2106.12015
openalex publication_date 2021/06/22 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We establish an asymptotic formula for the number of lattice points in the sets \mathbf Sh1, h2, h3(λ): =\x∈\mathbb Z+3:\lfloor h1(x1)\rfloor+\lfloor h2(x2)\rfloor+\lfloor h3(x3)\rfloor=λ\ with λ∈\mathbb 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 \mathbf Sc3(λ) := \x ∈ \mathbb Z3 : \lfloor |x1|c \rfloor + \lfloor |x2|c \rfloor + \lfloor |x3|c \rfloor= λ\ with λ∈\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 \frac1#\mathbf Sc3(λ)∑_n∈ \mathbf Sc3(λ)f(T1n1T2n2T3n3x) as λ→∞; 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>(11-4c)/(11-7c). Finally, we will study the equidistribution problem corresponding to the spheres \mathbf Sc3(λ).