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

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) an...

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Autores: Iosevich, A., Langowski, B., Mirek, M., Szarek, T.Z.
Formato: artículo
Estado:Versión publicada
Fecha de publicación:2023
País:España
Recursos:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/1635
Acesso em linha:http://hdl.handle.net/20.500.11824/1635
Access Level:acceso abierto
Palavra-chave:Lattice points, ergodic theorem, spherical maximal function, equidistribution problem, discrepancy, Fourier transform estimate, variational estimate, exponential sum
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spelling Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheresIosevich, A.Langowski, B.Mirek, M.Szarek, T.Z.Lattice points, ergodic theorem, spherical maximal function, equidistribution problem, discrepancy, Fourier transform estimate, variational estimate, exponential sumWe 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).Foundation for Polish Science via the START Scholarship, the Juan de la Cierva Incorporaci´on 2019, grant number IJC2019-039661-I, the Agencia Estatal de Investigaci´on, grant PID2020-113156GB-I00/AEI/10.13039/501100011033, the Basque Government through the BERC 2022-2025 program, and by the Spanish Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.202320232023info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/1635reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Inglésinfo:eu-repo/grantAgreement/MINECO//SEV-2017-0718info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113156GB-I00Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/16352026-06-19T12:47:47Z
dc.title.none.fl_str_mv Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
title Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
spellingShingle Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
Iosevich, A.
Lattice points, ergodic theorem, spherical maximal function, equidistribution problem, discrepancy, Fourier transform estimate, variational estimate, exponential sum
title_short Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
title_full Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
title_fullStr Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
title_full_unstemmed Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
title_sort Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres
dc.creator.none.fl_str_mv Iosevich, A.
Langowski, B.
Mirek, M.
Szarek, T.Z.
author Iosevich, A.
author_facet Iosevich, A.
Langowski, B.
Mirek, M.
Szarek, T.Z.
author_role author
author2 Langowski, B.
Mirek, M.
Szarek, T.Z.
author2_role author
author
author
dc.subject.none.fl_str_mv Lattice points, ergodic theorem, spherical maximal function, equidistribution problem, discrepancy, Fourier transform estimate, variational estimate, exponential sum
topic Lattice points, ergodic theorem, spherical maximal function, equidistribution problem, discrepancy, Fourier transform estimate, variational estimate, exponential sum
description 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).
publishDate 2023
dc.date.none.fl_str_mv 2023
2023
2023
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/1635
url http://hdl.handle.net/20.500.11824/1635
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/grantAgreement/MINECO//SEV-2017-0718
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2017-2020/PID2020-113156GB-I00
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
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dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
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collection BIRD. BCAM's Institutional Repository Data
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