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 tidsskrift › Tidsskriftartikel › fagfællebedømt
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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