Non-Euclidean visibility problems

  • Fernando Chamizo
Regular Articles


We consider the analog of visibility problems in hyperbolic plane (represented by Poincaré half-plane model ℍ), replacing the standard lattice ℤ × ℤ by the orbitz = i under the full modular group SL2(ℤ). We prove a visibility criterion and study orchard problem and the cardinality of visible points in large circles.


Modular group hyperbolic plane Poincaré half-plane model 


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  1. [1]
    Allen T T, On the arithmetic of phase locking: Coupled neurons as a lattice on ℝ2,Phys. D6 (1983) 305–320Google Scholar
  2. [2]
    Allen T T, Pólya’s orchard problem,Am. Math. Monthly 93 (1986) 98–104zbMATHCrossRefGoogle Scholar
  3. [3]
    Baake M, Moody R V and Pleasants P A B, Diffraction from visible lattice points and kth power free integers.Discrete Math. 221 (2000) 3–42zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Chamizo F, Some applications of large sieve in Riemann surfaces,Acta Arith. 77 (1996) 315–337MathSciNetGoogle Scholar
  5. [5]
    Erdös P, Gruber P M and Hammer J, Lattice points. Pitman Monographs and Surveys in Pure and Applied Mathematics, 39. Longman Scientific & Technical (1989)Google Scholar
  6. [6]
    Ellison W J, Les nombres premiers. En collaboration avec Michel Mendès France (Paris: Hermann) (1975)Google Scholar
  7. [7]
    Gauss C F, Disquisitiones arithmeticae (New York: Springer-Verlag) (1986)zbMATHGoogle Scholar
  8. [8]
    Heath-Brown D R, The distribution and moments of the error term in the Dirichlet divisor problem,Acta Arith. 60 (1992) 389–415zbMATHMathSciNetGoogle Scholar
  9. [9]
    Huxley M N, Introduction to Kloostermania. Elementary and analytic theory of numbers, Banach Center Publ., 17 (Warsaw: PWN) (1985) pp. 217–306Google Scholar
  10. [10]
    Iwaniec H, Introduction to the spectral theory of automorphic forms. Biblioteca de la Revista Matemática Iberoamericana. Revista Matemática Iberoamericana (1995)Google Scholar
  11. [11]
    Lovasz L, Pelikán J and Vesztergombi K, Discrete mathematics. Elementary and beyond. Undergraduate Texts in Mathematics (New York: Springer-Verlag) (2003)Google Scholar
  12. [12]
    Nowak W G, Primitive lattice points in rational ellipses and related arithmetic functions,Monatsh. Math. 106 (1988) 57–63zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    Phillips R and Rudnick Z, The circle problem in the hyperbolic plane,J. Funct. Anal. 121 (1994) 78–116zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    Polya G and Szegö G, Problems and theorems in analysis,II. Theory of functions, zeros, polynomials, determinants, number theory, geometry (New York, Heidelberg: Springer-Verlag) (1976)Google Scholar
  15. [15]
    Wesson P S, Valle K and Stabell R, The extragalactic background light and a definitive resolution of Olbers’s paradox,Astrophys. J. 317 (1987) 601–606CrossRefMathSciNetGoogle Scholar
  16. [16]
    Wu J, On the primitive circle problem,Monatsh. Math. 135 (2002) 69–81zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Indian Academy of Sciences 2006

Authors and Affiliations

  • Fernando Chamizo
    • 1
  1. 1.Departamento de Matemáticas, Facultad de CienciasUniversidad Autónoma de MadridMadridSpain

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