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

TL;DR

This paper proves a functional central limit theorem for the error term in the generalized circle problem for high-dimensional random lattices, showing convergence to Brownian motion under certain conditions.

Contribution

It introduces a new approach using a modified Rogers' mean value formula to analyze the error term's distribution in high dimensions.

Findings

Error term converges to Brownian motion in distribution

Develops a new version of Rogers' mean value formula

Establishes Gaussian moment convergence for certain functions

Abstract

In this note we study the error term R_{n,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(n) from the positive integers to the positive real line, tending to infinity with n but with subexponential growth. Then, the random function t -> (2f(n))^{-1/2} R_{n,L}(t f(n)) on the interval [0,1] converges in distribution to one-dimensional Brownian motion as n tends to infinity. The proof goes via convergence of moments, and for the computations we develop a new version of Rogers' mean value formula. For the individual k:th moment of the variable (2f(n))^{-1/2} R_{n,L}(f(n)) we prove convergence to the corresponding Gaussian moment more generally for functions f satisfying f(n)<<e^{cn} for any fixed c in an interval 0<c<c_k, where c_k is a constant depending on k whose optimal value we determine.