Functiones et Approximatio Commentarii Mathematici

A parity problem on the free path length of a billiard in the unit square with pockets

Andrew H. Ledoan, Alexandru Zaharescu, and Emre Alkan

Full-text: Open access

Abstract

We present a result on short intervals about the moments of the free path length of the linear trajectory of a billiard in the unit square with small triangular pockets of size $\varepsilon$ removed at the corners, in which the trajectory ends in a specified corner pocket.

Article information

Source
Funct. Approx. Comment. Math. Volume 35, Number 1 (2006), 19-36.

Dates
First available in Project Euclid: 16 December 2008

Permanent link to this document
http://projecteuclid.org/euclid.facm/1229442614

Mathematical Reviews number (MathSciNet)
MR2271604

Zentralblatt MATH identifier
1128.11042

Digital Object Identifier
doi:10.7169/facm/1229442614

Subjects
Primary: 37D50: Hyperbolic systems with singularities (billiards, etc.)
Secondary: 11B57: Farey sequences; the sequences ${1^k, 2^k, \cdots}$ 11P21: Lattice points in specified regions 82C40: Kinetic theory of gases

Keywords
Billiards periodic Lorentz gas free path length Farey fractions visible points Kloosterman sums

Citation

Alkan, Emre; Ledoan, Andrew H.; Zaharescu, Alexandru. A parity problem on the free path length of a billiard in the unit square with pockets. Functiones et Approximatio Commentarii Mathematici 35 (2006), no. 1, 19--36. doi:10.7169/facm/1229442614. http://projecteuclid.org/euclid.facm/1229442614.


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References

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  • F. P. Boca, C. Cobeli, and A. Zaharescu, A conjecture of R. R. Hall on Farey points, J. Reine Angew. Math. 535 (2001), 207--236.
  • F. P. Boca, C. Cobeli, and A. Zaharescu, On the distribution of the Farey sequence with odd denominators, Michigan Math. J. 51 (2003), no. 3, 557--574.
  • F. P. Boca, R. N. Gologan, and 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.
  • F. P. Boca, R. N. Gologan, and A. Zaharescu, The statistics of the trajectory of a certain billiard in a flat two-torus, Commun. Math. Phys. 240 (2003), no. 1-2, 53--73.
  • F. P. Boca and A. Zaharescu, On the correlations of directions in the Euclidean plane, Trans. Amer. Math. Soc. 358 (2006), 1797--1825.
  • F. P. Boca and A. Zaharescu, Farey fractions and two-dimentional tori, in Noncommutative Geometry and Number Theory (C. Consani, M. Marcolli, eds.), Aspects of Mathematics E37, Vieweg Verlag, Wiesbaden, 2006, 57--77.
  • J. Bourgain, F. Golse, and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas, Commun. Math. Phys. 190 (1998), 491--508.
  • L. A. Bunimovich, Billiards and other hyperbolic systems, in Dynamical systems, ergodic theory and applications (Ya. G. Sinai and al. eds, translated from the Russian), 192--233, Encyclopaedia of Math. Sci. 100, 2nd ed., Springer-Verlag, Berlin, 2000.
  • E. Caglioti and F. Golse, On the distribution of free path lengths for the periodic Lorentz gas. III, Commun. Math. Phys. 236 (2003), no. 2, 199--221.
  • C. Cobeli and A. Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform, no. 5 (2003), 1--38.
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  • T. Estermann, On Kloosterman's sum, Mathematika 8 (1961), 83--86.
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  • R. R. Hall, A note on Farey series, J. London Math. Soc. 2 (1970), pp. 139--148.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1979.
  • A. Haynes, A note on Farey fractions with odd denominators, J. Number Theory 98 (2003), no. 1, 89--104.
  • A. Haynes, The distribution of special subsets of the Farey sequence, J. Number Theory 107 (2004), no. 1, 95--104.
  • C. Hooley, An asymptotic formula in the theory of numbers, Proc. London Math. Soc. 7 (1957), 396--413.
  • M. N. Huxley, The distribution of Farey points. I, Acta. Arith. 18 (1971), 281--287.
  • H. A. Lorentz, Le mouvement des électrons dans les métaux, Arch. Neerl. 10 (1905), 336; reprinted in Collected papers Vol. 3, Martinus Nijhoff, The Hague, 1936, 180--214.
  • A. Weil, On some exponential sums, Proc. Nat. Acad. U. S. A. 34 (1948), 204--207.
  • F. P. Boca, C. Cobeli, and A. Zaharescu, Distribution of lattice points visible from the origin, Commun. Math. Phys. 213 (2000), 433--470.
  • F. P. Boca, C. Cobeli, and A. Zaharescu, A conjecture of R. R. Hall on Farey points, J. Reine Angew. Math. 535 (2001), 207--236.
  • F. P. Boca, C. Cobeli, and A. Zaharescu, On the distribution of the Farey sequence with odd denominators, Michigan Math. J. 51 (2003), no. 3, 557--574.
  • F. P. Boca, R. N. Gologan, and 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.
  • F. P. Boca, R. N. Gologan, and A. Zaharescu, The statistics of the trajectory of a certain billiard in a flat two-torus, Commun. Math. Phys. 240 (2003), no. 1-2, 53--73.
  • F. P. Boca and A. Zaharescu, On the correlations of directions in the Euclidean plane, Trans. Amer. Math. Soc. 358 (2006), 1797--1825.
  • F. P. Boca and A. Zaharescu, Farey fractions and two-dimentional tori, in Noncommutative Geometry and Number Theory (C. Consani, M. Marcolli, eds.), Aspects of Mathematics E37, Vieweg Verlag, Wiesbaden, 2006, 57--77.
  • J. Bourgain, F. Golse, and B. Wennberg, On the distribution of free path lengths for the periodic Lorentz gas, Commun. Math. Phys. 190 (1998), 491--508.
  • L. A. Bunimovich, Billiards and other hyperbolic systems, in Dynamical systems, ergodic theory and applications (Ya. G. Sinai and al. eds, translated from the Russian), 192--233, Encyclopaedia of Math. Sci. 100, 2nd ed., Springer-Verlag, Berlin, 2000.
  • E. Caglioti and F. Golse, On the distribution of free path lengths for the periodic Lorentz gas. III, Commun. Math. Phys. 236 (2003), no. 2, 199--221.
  • C. Cobeli and A. Zaharescu, The Haros-Farey sequence at two hundred years, Acta Univ. Apulensis Math. Inform, no. 5 (2003), 1--38.
  • H. S. M. Coxeter, Introduction to Geometry, John Wiley & Sons, 1961, New York.
  • T. Estermann, On Kloosterman's sum, Mathematika 8 (1961), 83--86.
  • A. Fujii, On a problem of Dinaburg and Sinaĭ, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 7, 198--203.
  • R. R. Hall, A note on Farey series, J. London Math. Soc. 2 (1970), pp. 139--148.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Clarendon Press, Oxford, 1979.
  • A. Haynes, A note on Farey fractions with odd denominators, J. Number Theory 98 (2003), no. 1, 89--104.
  • A. Haynes, The distribution of special subsets of the Farey sequence, J. Number Theory 107 (2004), no. 1, 95--104.
  • C. Hooley, An asymptotic formula in the theory of numbers, Proc. London Math. Soc. 7 (1957), 396--413.
  • M. N. Huxley, The distribution of Farey points. I, Acta. Arith. 18 (1971), 281--287.
  • H. A. Lorentz, Le mouvement des électrons dans les métaux, Arch. Neerl. 10 (1905), 336; reprinted in Collected papers Vol. 3, Martinus Nijhoff, The Hague, 1936, 180--214.
  • A. Weil, On some exponential sums, Proc. Nat. Acad. U. S. A. 34 (1948), 204--207.