Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the correlations of directions in the Euclidean plane


Authors: Florin P. Boca and Alexandru Zaharescu
Journal: Trans. Amer. Math. Soc. 358 (2006), 1797-1825
MSC (2000): Primary 11J71; Secondary 11J20, 11P21
Published electronically: October 21, 2005
MathSciNet review: 2186997
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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{displaymath}\frac{1}{\operatorname{Area} ({\mathbb{D}}_{0})} \iint \limi... ...1}{10}+\delta }) \qquad \text{\rm as $Q\rightarrow \infty $ .} \end{displaymath}

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 11J71, 11J20, 11P21

Retrieve articles in all journals with MSC (2000): 11J71, 11J20, 11P21


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
Email: zaharesc@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03783-9
PII: S 0002-9947(05)03783-9
Keywords: Directions in ${\mathbb{R}}^{2}$, correlation measures
Received by editor(s): May 4, 2004
Received by editor(s) in revised form: July 9, 2004
Published electronically: October 21, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.