From exponential counting to pair correlations

We prove an abstract result on the correlations of pairs of elements in an exponentially growing discrete subset E of [0,+ infinity[ endowed with a weight function. Assume that there exist alpha is an element of 2 R, c, delta > 0 such that, as t -> +infinity, the weighted number (w) over tilde (t) of elements of E that are not greater than t is equivalent to ct(alpha)e(delta t). We prove that the distribution function of the differences of elements of E is t bar right arrow delta/2 e(-vertical bar t vertical bar), and that, under an error term assumption on (w) over tilde (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 lengths of closed geodesics and common perpendiculars in negatively curved manifolds and metric graphs.