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,y)P, where P _(x,Y) is the fixed point (x, y) ∈ [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 double-struck D sign ₀ converges as Q → ∞. More precisely we prove, for every λ ∈ ℝ ₊ and 0 < δ < 1/10, the estimate 1/Area(double-struck D sign ₀) ∫∫D ₀ R _(x,y),Q ⁽²⁾(λ)dx dy = 2πλ/3 + Odouble-struck D sign _0,λ,δ(Q ^-1/10+δ) as Q → ∞. We also proves that for each individual point (x, y) ∈ [0, 1) ², the 6-level correlation R _(x,y),Q ⁽⁶⁾ (λ) diverges at any point λ ∈ ℝ ₊ ⁵ as Q → ∞, and we give an explicit lower bound for the rate of divergence.