On the correlations of directions in the Euclidean plane

Let R-(x,y),Q((v)) denote the repartition of the v-level correlation measure of the finite set of directions P-(x,P- y) P, where P-(x,P- y) is the fixed point (x, y) is an element of [0, 1](2) and P is an integer lattice point in the square [-Q, Q](2). We show that the average of the pair correlation repartition R-(x,y),Q((2)) over (x, y) in a fixed disc D-0 converges as Q -> infinity. More precisely we prove, for every lambda is an element of R+ and 0 < delta < 1/10, the estimate 1/Area(D-0) integral integral R-Do((x,y),Q)(2)(lambda)dxdy = 2 pi lambda/3 + O-D0,(lambda),delta(Q(-1/10+delta)) as Q -> infinity. We also prove that for each individual point (x,y) is an element of [0, 1](2), the 6-level correlation R-(x,y),Q((6))(lambda) diverges at any point lambda is an element of R-+(5) as Q -> infinity, and we give an explicit lower bound for the rate of divergence.