On the correlations of directions in the Euclidean plane

Abstract

Let R (v) (x,y),Q 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 (2) (x,y),Q over (x, y) in a fixed disc D 0 converges as Q → oo. More precisely we prove, for every A ∈ R + and 0 < δ < 1/10, the estimate formula math. We also prove that for each individual point (x,y) ∈ [0,1) 2 , the 6-level correlation R (6) (x,y),Q (λ) diverges at any point A ∈ R 5 + as Q → ∞, and we give an explicit lower bound for the rate of divergence.