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
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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.
Originalsprog | Engelsk |
---|---|
Tidsskrift | Advances in Mathematics |
Vol/bind | 345 |
Sider (fra-til) | 1042-1074 |
Antal sider | 33 |
ISSN | 0001-8708 |
DOI | |
Status | Udgivet - 17 mar. 2019 |
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