On the correlations of directions in the Euclidean plane
Authors:
Florin P. Boca and Alexandru Zaharescu
Journal:
Trans. Amer. Math. Soc. 358 (2006), 17971825
MSC (2000):
Primary 11J71; Secondary 11J20, 11P21
Published electronically:
October 21, 2005
MathSciNet review:
2186997
Fulltext PDF Free Access
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.
 [1]
Volker
Augustin, Florin
P. Boca, Cristian
Cobeli, and Alexandru
Zaharescu, The ℎspacing distribution between Farey
points, Math. Proc. Cambridge Philos. Soc. 131
(2001), no. 1, 23–38. MR 1833071
(2002h:11017), http://dx.doi.org/10.1017/S0305004101005187
 [2]
Florin
P. Boca, Cristian
Cobeli, and Alexandru
Zaharescu, Distribution of lattice points visible from the
origin, Comm. Math. Phys. 213 (2000), no. 2,
433–470. MR 1785463
(2001j:11094), http://dx.doi.org/10.1007/s002200000250
 [3]
Florin
P. Boca and Alexandru
Zaharescu, Pair correlation of values of rational functions (mod
𝑝), Duke Math. J. 105 (2000), no. 2,
267–307. MR 1793613
(2001j:11065), http://dx.doi.org/10.1215/S0012709400105248
 [4]
F.
P. Boca, R.
N. Gologan, and A.
Zaharescu, The average length of a trajectory in a certain billiard
in a flat twotorus, New York J. Math. 9 (2003),
303–330 (electronic). MR 2028172
(2004m:37065)
 [5]
F. P. Boca, A. Zaharescu, The correlations of Farey fractions, to appear in J. London Math. Soc.
 [6]
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R. Hall, A note on Farey series, J. London Math. Soc. (2)
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(91i:11001)
 [8]
Zeév
Rudnick and Peter
Sarnak, The pair correlation function of fractional parts of
polynomials, Comm. Math. Phys. 194 (1998),
no. 1, 61–70. MR 1628282
(99g:11088), http://dx.doi.org/10.1007/s002200050348
 [9]
Zeév
Rudnick, Peter
Sarnak, and Alexandru
Zaharescu, The distribution of spacings between the fractional
parts of 𝑛²𝛼, Invent. Math. 145
(2001), no. 1, 37–57. MR 1839285
(2002e:11093), http://dx.doi.org/10.1007/s002220100141
 [10]
Alexandru
Zaharescu, Correlation of fractional parts of
𝑛²𝛼, Forum Math. 15 (2003),
no. 1, 1–21. MR 1957276
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 [1]
 V. Augustin, F.P. Boca, C. Cobeli, A. Zaharescu, The spacing distribution between Farey points, Math. Proc. Camb. Phil. Soc. 131 (2001), 2338. MR 1833071 (2002h:11017)
 [2]
 F. P. Boca, C. Cobeli, A. Zaharescu, Distribution of lattice points visible from the origin, Comm. Math. Phys. 213 (2000), 433470. MR 1785463 (2001j:11094)
 [3]
 F. P. Boca, A. Zaharescu, Pair correlation of values of rational functions, Duke Math. J. 105 (2000), 267307. MR 1793613 (2001j:11065)
 [4]
 F. P. Boca, R. N. Gologan, A. Zaharescu, The average length of a trajectory in a certain billiard in a flat twotorus, New York J. Math. 9 (2003), 303330. MR 2028172 (2004m:37065)
 [5]
 F. P. Boca, A. Zaharescu, The correlations of Farey fractions, to appear in J. London Math. Soc.
 [6]
 R. R. Hall, A note on Farey series, J. London Math. Soc. 2 (1970), 139148. MR 0253978 (40:7191)
 [7]
 I. Niven, H. S. Zuckerman, H. L. Montgomery, An introduction to the theory of numbers, John Wiley & Sons, Inc., 1991. MR 1083765 (91i:11001)
 [8]
 Z. Rudnick, P. Sarnak, The pair correlation function of fractional parts of polynomials, Comm. Math. Phys. 194 (1998), 6170. MR 1628282 (99g:11088)
 [9]
 Z. Rudnick, P. Sarnak, A. Zaharescu, The distribution of spacings between the fractional parts of , Invent. Math. 145 (2001), 3757. MR 1839285 (2002e:11093)
 [10]
 A. Zaharescu, Correlation of fractional parts of , Forum Math. 15 (2003), 121. MR 1957276 (2004a:11065)
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Additional Information
Florin P. Boca
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 West Green Street, Urbana, Illinois 61801
Email:
fboca@math.uiuc.edu
Alexandru Zaharescu
Affiliation:
Department of Mathematics, University of Illinois at UrbanaChampaign, 1409 West Green Street, Urbana, Illinois 61801
Email:
zaharesc@math.uiuc.edu
DOI:
http://dx.doi.org/10.1090/S0002994705037839
PII:
S 00029947(05)037839
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.
