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

Publikation: Bidrag til tidsskriftTidsskriftartikelForskningfagfællebedømt

Andreas Strömbergsson, Anders Södergren

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 Advances in Mathematics 345 1042-1074 33 0001-8708 https://doi.org/10.1016/j.aim.2019.01.034 Udgivet - 17 mar. 2019

ID: 212678831