American Institute of Mathematical Sciences

Advanced
January  2016, 15(1): 1-8. doi: 10.3934/cpaa.2016.15.1

Average number of lattice points in a disk

Sujay Jayakar 1, and  Robert S. Strichartz 2,

1. 

3633 19th St., San Francisco, CA 94110, United States
2. 

Mathematics Department, Malott Hall, Cornell Univeristy, Ithaca, NY 14853, United States

Received  November 2014 Revised  October 2015 Published  December 2015

The difference between the number of lattice points in a disk of radius $\sqrt{t}/2\pi$ and the area of the disk $t/4\pi$ is equal to the error in the Weyl asymptotic estimate for the eigenvalue counting function of the Laplacian on the standard flat torus. We give a sharp asymptotic expression for the average value of the difference over the interval $0 \leq t \leq R$. We obtain similar results for families of ellipses. We also obtain relations to the eigenvalue counting function for the Klein bottle and projective plane.
Keywords: Bessel function., Lattice points, Weyl asymptotics.
Mathematics Subject Classification: Primary: 35J05; Secondary: 42B9.
Citation: Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure & Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1
References:
[1]

M. Begué, T. Kalloniatis and R. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet,, \emph{Fractals}, 21 (2013). doi: 10.1142/S0218348X13500023. Google Scholar

[2]

P. Bleher, On the distribution of the number of lattice points inside a family of convex ovals,, \emph{Duke Math J.}, (): 461. doi: 10.1215/S0012-7094-92-06718-4. Google Scholar

[3]

P. Bleher, Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems,, \emph{Duke Math J.}, (): 655. doi: 10.1215/S0012-7094-93-07015-9. Google Scholar

[4]

M. Huxley, The mean lattice discrepancy,, \emph{Proc. Edinburg Math. Soc.}, 38 (1995), 523. doi: 10.1017/S0013091500019313. Google Scholar

[5]

H. Iwaniec and E. Kowalski, Analytic Number Theory,, AMS Colloq. Publ. vol 53, (2004). Google Scholar

[6]

S. Jayakar and R. Strichartz, Average number of lattice points in a disk,, \url{http://www.math.cornell.edu/ sujay/lattice}, (2012). Google Scholar

[7]

N. Lebedev, Special Functions and Their Applications,, Dover Publications, (1965). Google Scholar

[8]

W. Müller, On the average order of the lattice rest of a convex body,, \emph{Acta Arith.}, 80 (1997), 89. Google Scholar

[9]

C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian,, Princeton Univ. Press, (2014). doi: 10.1515/9781400850549. Google Scholar

[10]

E. Stein and R. Shakarchi, Functional Analysis,, Princeton Univ. Press, (2011). Google Scholar

[11]

R. Strichartz, Average error for spectral asymptotics on surfaces,, \emph{Comm. Pure Appl. Analysis}, 15 (2016), 9. Google Scholar

show all references

References:
[1]

M. Begué, T. Kalloniatis and R. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet,, \emph{Fractals}, 21 (2013). doi: 10.1142/S0218348X13500023. Google Scholar

[2]

P. Bleher, On the distribution of the number of lattice points inside a family of convex ovals,, \emph{Duke Math J.}, (): 461. doi: 10.1215/S0012-7094-92-06718-4. Google Scholar

[3]

P. Bleher, Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems,, \emph{Duke Math J.}, (): 655. doi: 10.1215/S0012-7094-93-07015-9. Google Scholar

[4]

M. Huxley, The mean lattice discrepancy,, \emph{Proc. Edinburg Math. Soc.}, 38 (1995), 523. doi: 10.1017/S0013091500019313. Google Scholar

[5]

H. Iwaniec and E. Kowalski, Analytic Number Theory,, AMS Colloq. Publ. vol 53, (2004). Google Scholar

[6]

S. Jayakar and R. Strichartz, Average number of lattice points in a disk,, \url{http://www.math.cornell.edu/ sujay/lattice}, (2012). Google Scholar

[7]

N. Lebedev, Special Functions and Their Applications,, Dover Publications, (1965). Google Scholar

[8]

W. Müller, On the average order of the lattice rest of a convex body,, \emph{Acta Arith.}, 80 (1997), 89. Google Scholar

[9]

C. Sogge, Hangzhou Lectures on Eigenfunctions of the Laplacian,, Princeton Univ. Press, (2014). doi: 10.1515/9781400850549. Google Scholar

[10]

E. Stein and R. Shakarchi, Functional Analysis,, Princeton Univ. Press, (2011). Google Scholar

[11]

R. Strichartz, Average error for spectral asymptotics on surfaces,, \emph{Comm. Pure Appl. Analysis}, 15 (2016), 9. Google Scholar

[1]

Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
[2]

Ha Pham, Plamen Stefanov. Weyl asymptotics of the transmission eigenvalues for a constant index of refraction. Inverse Problems & Imaging, 2014, 8 (3) : 795-810. doi: 10.3934/ipi.2014.8.795
[3]

Gerald Teschl. On the spatial asymptotics of solutions of the Toda lattice. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1233-1239. doi: 10.3934/dcds.2010.27.1233
[4]

Dmitry Jakobson and Iosif Polterovich. Lower bounds for the spectral function and for the remainder in local Weyl's law on manifolds. Electronic Research Announcements, 2005, 11: 71-77.

[5]

Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975
[6]

Dmitry Kleinbock, Xi Zhao. An application of lattice points counting to shrinking target problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 155-168. doi: 10.3934/dcds.2018007
[7]

Dubi Kelmer. Approximation of points in the plane by generic lattice orbits. Journal of Modern Dynamics, 2017, 11: 143-153. doi: 10.3934/jmd.2017007
[8]

Zhaoquan Xu, Jiying Ma. Monotonicity, asymptotics and uniqueness of travelling wave solution of a non-local delayed lattice dynamical system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5107-5131. doi: 10.3934/dcds.2015.35.5107
[9]

Uri Bader, Alex Furman. Boundaries, Weyl groups, and Superrigidity. Electronic Research Announcements, 2012, 19: 41-48. doi: 10.3934/era.2012.19.41
[10]

M.T. Boudjelkha. Extended Riemann Bessel functions. Conference Publications, 2005, 2005 (Special) : 121-130. doi: 10.3934/proc.2005.2005.121
[11]

Frédéric Naud, Anke Pohl, Louis Soares. Fractal Weyl bounds and Hecke triangle groups. Electronic Research Announcements, 2019, 26: 24-35. doi: 10.3934/era.2019.26.003
[12]

Xiaolong Han, Guozhen Lu. Regularity of solutions to an integral equation associated with Bessel potential. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1111-1119. doi: 10.3934/cpaa.2011.10.1111
[13]

Yutian Lei. Positive solutions of integral systems involving Bessel potentials. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2721-2737. doi: 10.3934/cpaa.2013.12.2721
[14]

Jesse Goodman, Daniel Spector. Some remarks on boundary operators of Bessel extensions. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 493-509. doi: 10.3934/dcdss.2018027
[15]

Yonggang Zhao, Mingxin Wang. An integral equation involving Bessel potentials on half space. Communications on Pure & Applied Analysis, 2015, 14 (2) : 527-548. doi: 10.3934/cpaa.2015.14.527
[16]

Mingchun Wang, Jiankai Xu, Huoxiong Wu. On Positive solutions of integral equations with the weighted Bessel potentials. Communications on Pure & Applied Analysis, 2019, 18 (2) : 625-641. doi: 10.3934/cpaa.2019031
[17]

Kurt Vinhage. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. Journal of Modern Dynamics, 2015, 9: 25-49. doi: 10.3934/jmd.2015.9.25
[18]

Lu Chen, Zhao Liu, Guozhen Lu. Qualitative properties of solutions to an integral system associated with the Bessel potential. Communications on Pure & Applied Analysis, 2016, 15 (3) : 893-906. doi: 10.3934/cpaa.2016.15.893
[19]

Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3103-3131. doi: 10.3934/dcds.2015.35.3103
[20]

Magnus Aspenberg, Fredrik Ekström, Tomas Persson, Jörg Schmeling. On the asymptotics of the scenery flow. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2797-2815. doi: 10.3934/dcds.2015.35.2797

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences