On the correlations of directions in the Euclidean plane

Let ${\mathcal{R}}^{(\nu )}_{(x,y),Q}$ denote the repartition of the $\nu$-level correlation measure of the finite set of directions $P_{(x,y)}P$, where $P_{(x,y)}$ is the fixed point $(x,y)\in [0,1)^{2}$ and $P$ is an integer lattice point in the square $[-Q,Q]^{2}$. We show that the average of the pair correlation repartition ${\mathcal{R}}^{(2)}_{(x,y),Q}$ over $(x,y)$ in a fixed disc ${\mathbb{D}}_{0}$ converges as $Q\rightarrow \infty$. More precisely we prove, for every $\lambda \in {\mathbb{R}}_{+}$ and $0<\delta <\frac{1}{10}$, the estimate

We also prove that for each individual point $(x,y)\in [0,1)^{2}$, the $6$-level correlation ${\mathcal{R}}^{(6)}_{(x,y),Q}(\lambda )$diverges at any point $\lambda \in {\mathbb{R}}^{5}_{+}$ as $Q\rightarrow \infty$, and we give an explicit lower bound for the rate of divergence.