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Transactions of the American Mathematical Society

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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 ${\mathcal{R}}^{(\nu )}_{(x,y),Q}$ 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?)

  • [1] V. Augustin, F.P. Boca, C. Cobeli, A. Zaharescu, The $h$-spacing distribution between Farey points, Math. Proc. Camb. Phil. Soc. 131 (2001), 23-38. 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), 433-470. MR 1785463 (2001j:11094)
  • [3] F. P. Boca, A. Zaharescu, Pair correlation of values of rational functions$\pmod {p}$, Duke Math. J. 105 (2000), 267-307. 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 two-torus, New York J. Math. 9 (2003), 303-330. 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), 139-148. 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), 61-70. MR 1628282 (99g:11088)
  • [9] Z. Rudnick, P. Sarnak, A. Zaharescu, The distribution of spacings between the fractional parts of $n^{2} \alpha $, Invent. Math. 145 (2001), 37-57. MR 1839285 (2002e:11093)
  • [10] A. Zaharescu, Correlation of fractional parts of $n^{2} \alpha $, Forum Math. 15 (2003), 1-21. MR 1957276 (2004a:11065)

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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.

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