2022/01/28 by Parkkonen, Jouni, Paulin, Frédéric
#05A16 #11N45 #26E99 #28A33 #37C35 #53C22 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2201.12118
We prove an abstract result on the correlations of pairs of elements in an exponentially growing discrete subset \mathcal E of [0,+∞[ endowed with a weight function. Assume that there exist α∈\mathbb R, c,δ>0 such that, as t→+∞, the weighted number \widetildeω(t) of elements of \mathcal E that are not greater than t is equivalent to c tαeδt. We prove that the distribution function of the unscaled differences of elements of \mathcal E is t↦\fracδ2 e-|t|, and that, under an error term assumption on \widetildeω(t), the pair correlation with a scaling with polynomial growth exhibits a Poissonian behaviour. We apply this result to answer a question of Pollicott and Sharp on the pair correlations of closed geodesics and common perpendiculars in negatively curved manifolds and metric graphs.