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

Journal article, 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.