2020/10/02 by Gath, Yoav A.
#FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)
paper · doi:10.48550/arxiv.2010.01096
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 Eq(x)=|ℤ2q+1∩δxB|-vol(B)x2q+2 denote the error term which occurs for this lattice point counting problem on the Heisenberg group ℍq, where B is the unit ball in the Cygan-Korányi norm and δx is the Heisenberg-dilation by x>0. For q≥3 we consider the suitably normalized error term Eq(x)/x2q-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 Pq(α), can be extended to the whole complex plane ℂ as an entire function of α and satisfies for any non-negative integer j≥0 and any α∈ℝ, |α|>αq,j, the bound: \beginequation* \beginsplit |P(j)q(α)|≤exp(-|α|^4-β/loglog|α|) split equation* where β>0 is an absolute constant. In addition, we give an explicit formula for the j-th integral moment of the density Pq(α) for any integer j≥1.