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
Accepted author manuscript, 360 KB, PDF document
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.
Original language | English |
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Journal | Advances in Mathematics |
Volume | 345 |
Pages (from-to) | 1042-1074 |
Number of pages | 33 |
ISSN | 0001-8708 |
DOIs | |
Publication status | Published - 17 Mar 2019 |
- Brownian motion, Random lattice, Rogers’ mean value formula, The generalized circle problem
Research areas
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