On the generalized circle problem for a random lattice in large dimension
Artikel i vetenskaplig tidskrift, 2019

In this note we study the error term R-n,R-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 lim(n ->infinity) f (n) = infinity and f (n) = O-epsilon(e(epsilon n)) for every epsilon > 0. Then, the random function t bar right arrow 1/root 2f (n) R-n,R-L (t f(n)) on the interval [0, 1] converges in distribution to one-dimensional Brownian motion as n -> infinity. 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))(-1/2) R-n,R-L (f (n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f (n) = O(e(cn)) for any fixed c is an element of (0, c(k)), where c(k) is a constant depending on k whose optimal value we determine. (C) 2019 Elsevier Inc. All rights reserved.

Brownian motion

The generalized circle problem

Random lattice

Rogers' mean value formula


Andreas Strombergsson

Uppsala universitet

Anders Södergren

Köpenhamns universitet

Advances in Mathematics

0001-8708 (ISSN) 1090-2082 (eISSN)

Vol. 345 1042-1074



Sannolikhetsteori och statistik

Matematisk analys



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