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

DOI:
https://doi.org/10.1090/S0002-9947-05-03783-9

Published electronically:
October 21, 2005

MathSciNet review:
2186997

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Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the repartition of the -level correlation measure of the finite set of directions , where is the fixed point and is an integer lattice point in the square . We show that the average of the pair correlation repartition over in a fixed disc converges as . More precisely we prove, for every and , the estimate

We also prove that for each individual point , the -level correlation diverges at any point as , and we give an explicit lower bound for the rate of divergence.

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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

Email:
zaharesc@math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03783-9

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.