On the generalized circle problem for a random lattice in large dimension

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On the generalized circle problem for a random lattice in large dimension. / Strömbergsson, Andreas; Södergren, Anders.

I: Advances in Mathematics, Bind 345, 17.03.2019, s. 1042-1074.

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

Harvard

Strömbergsson, A & Södergren, A 2019, 'On the generalized circle problem for a random lattice in large dimension', Advances in Mathematics, bind 345, s. 1042-1074. https://doi.org/10.1016/j.aim.2019.01.034

APA

Strömbergsson, A., & Södergren, A. (2019). On the generalized circle problem for a random lattice in large dimension. Advances in Mathematics, 345, 1042-1074. https://doi.org/10.1016/j.aim.2019.01.034

Vancouver

Strömbergsson A, Södergren A. On the generalized circle problem for a random lattice in large dimension. Advances in Mathematics. 2019 mar. 17;345:1042-1074. https://doi.org/10.1016/j.aim.2019.01.034

Author

Strömbergsson, Andreas ; Södergren, Anders. / On the generalized circle problem for a random lattice in large dimension. I: Advances in Mathematics. 2019 ; Bind 345. s. 1042-1074.

Bibtex

@article{ce7ba2409ebe4916862d429e52de2537,
title = "On the generalized circle problem for a random lattice in large dimension",
abstract = "In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z+→R+ satisfying limn→∞⁡f(n)=∞ and f(n)=Oε(eεn) for every ε>0. Then, the random function t↦[Formula presented]Rn,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers{\textquoteright} mean value formula from [18]. For the individual kth moment of the variable (2f(n))−1/2Rn,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(ecn) for any fixed c∈(0,ck), where ck is a constant depending on k whose optimal value we determine.",
keywords = "Brownian motion, Random lattice, Rogers{\textquoteright} mean value formula, The generalized circle problem",
author = "Andreas Str{\"o}mbergsson and Anders S{\"o}dergren",
year = "2019",
month = mar,
day = "17",
doi = "10.1016/j.aim.2019.01.034",
language = "English",
volume = "345",
pages = "1042--1074",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press",

}

RIS

TY - JOUR

T1 - On the generalized circle problem for a random lattice in large dimension

AU - Strömbergsson, Andreas

AU - Södergren, Anders

PY - 2019/3/17

Y1 - 2019/3/17

N2 - In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z+→R+ satisfying limn→∞⁡f(n)=∞ and f(n)=Oε(eεn) for every ε>0. Then, the random function t↦[Formula presented]Rn,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers’ mean value formula from [18]. For the individual kth moment of the variable (2f(n))−1/2Rn,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(ecn) for any fixed c∈(0,ck), where ck is a constant depending on k whose optimal value we determine.

AB - In this note we study the error term Rn,L(x) in the generalized circle problem for a ball of volume x and a random lattice L of large dimension n. Our main result is the following functional central limit theorem: Fix an arbitrary function f:Z+→R+ satisfying limn→∞⁡f(n)=∞ and f(n)=Oε(eεn) for every ε>0. Then, the random function t↦[Formula presented]Rn,L(tf(n)) on the interval [0,1] converges in distribution to one- dimensional Brownian motion as n→∞. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers’ mean value formula from [18]. For the individual kth moment of the variable (2f(n))−1/2Rn,L(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)=O(ecn) for any fixed c∈(0,ck), where ck is a constant depending on k whose optimal value we determine.

KW - Brownian motion

KW - Random lattice

KW - Rogers’ mean value formula

KW - The generalized circle problem

UR - http://www.scopus.com/inward/record.url?scp=85060482231&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.01.034

DO - 10.1016/j.aim.2019.01.034

M3 - Journal article

AN - SCOPUS:85060482231

VL - 345

SP - 1042

EP - 1074

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -

ID: 212678831