## On the correlations of directions in the Euclidean plane

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- by Florin P. Boca and Alexandru Zaharescu PDF
- Trans. Amer. Math. Soc.
**358**(2006), 1797-1825 Request permission

## 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.## References

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## Additional Information

**Florin P. Boca**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
- Email: fboca@math.uiuc.edu
**Alexandru Zaharescu**- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
- MR Author ID: 186235
- Email: zaharesc@math.uiuc.edu
- Received by editor(s): May 4, 2004
- Received by editor(s) in revised form: July 9, 2004
- Published electronically: October 21, 2005
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**358**(2006), 1797-1825 - MSC (2000): Primary 11J71; Secondary 11J20, 11P21
- DOI: https://doi.org/10.1090/S0002-9947-05-03783-9
- MathSciNet review: 2186997