On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

Abstract

A. We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Korányi norm ball of large radius. Let E ( ) = Z 2 +1 ∩ B − vol B 2 +2 denote the error term which occurs for this lattice point counting problem on the Heisenberg group H , where B is the unit ball in the Cygan-Korányi norm and is the Heisenberg-dilation by > 0. The characteristic behavior of the error term E ( ) may only be one of two types, depending on whether = 1 or > 1. It is the higher dimensional case that is the most challenging, and we shall confine ourselves to the case ≥ 3. To that end, for ≥ 3 we consider the suitably normalized error term E ( )/ 2 −1 , and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by P ( ), can be extended to the whole complex plane C as an entire function of and satisfies for any non-negative integer ≥ 0 and any ∈ R, | | > , , the bound:where > 0 is an absolute constant. In addition, we prove that ∫ ∞ −∞ P ( )d = 0 and ∫ ∞ −∞ 3 P ( )d < 0, and we give an explicit formula for the -th integral moment of the density P ( ) for any integer ≥ 1. Finally, we show that for = 1, 2, the -th integral moment of E ( )/ 2 −1 is equal to ∫ ∞ −∞ P ( )d , and more generally: lim →∞ 1 2 ∫ E ( )/ 2 −1 d = ∞ ∫ −∞ | | P ( )d for any 0 < ≤ 2.

Distribution and Moments of the Error Term in the Lattice Point Counting Problem for 3-Dimensional Cygan-Korányi Balls

“…This is in sharp contrast to the higher dimensional case ≥ 3, where the corresponding functions ϕ , ( ) (see [11], §2 Theorem 4) are aperiodic. Also, the presence of the factor 1/ in ϕ 1, ( ), as apposed to 1/ 3/4 in ϕ , ( ) for ≥ 3, is the reason for why we obtain the much stronger decay estimates (1.2) compared to the higher dimensional case ≥ 3.…”

“…Our goal in the present paper is to investigate the nature in which 1 ( ) fluctuates around its expected value vol B 4 . In the higher dimensional case ≥ 3, this has been carried out by the author, and we have the following result ( [11], Theorem 1 and Theorem 3).…”

“…Theorem ( [11]). Let ≥ 3 be an integer, and let E ( ) = E ( )/ 2 −1 be the suitably normalized error term.…”