On the correlations of directions in the Euclidean plane

Abstract:

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 \begin{equation*} \frac {1}{\operatorname {Area} ({\mathbb {D}}_{0})} \iint _{\mathbb {D}_0} \mathcal {R}^{(2)}_{(x,y),Q} (\lambda )\, dx\, dy = \frac {2\pi \lambda }{3} + O_{\mathbb {D}_0, \lambda , \delta } (Q^{-\frac {1}{10}+\delta }) \;\; \text {as $Q\rightarrow \infty $.} \end{equation*} 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.