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Borel-Cantelli sequences

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Abstract

A sequence {x n } ∞1 in the unit interval [0, 1) = ℝ/ℤ is called Borel-Cantelli, or BC, if for all non-increasing sequences of positive real numbers {a n } with 48001013 1. \(\sum\nolimits_{i = 1}^\infty {} {a_i} = \infty \), the set

$$\{ x \in [0,1)\left| {\left| {x - {x_n}} \right|} \right. < {a_n}{\rm{for infinitely many }}n \ge 1\} $$

has full Lebesgue measure. (Speaking informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds).

The notion of BC sequences is motivated by the monotone shrinking target property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC sequences and sequences that are not BC are also presented.

The property of a sequence to be BC is a delicate Diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC, while the orbits of a “generic” IET are not.

The notion of BC sequences is extended from [0, 1) to sequences in Ahlfors regular spaces.

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References

  1. J. Athreya, Logarithm laws and shrinking target properties, Proc. Indian Acad. Sci. Math. Sci. 119 (2009), 541–557.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Baker and W. Schmidt, Diophantine approximation and Hausdorff dimension. Proc. London Math. Soc. (3) 21 (1970), 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  3. V. Beresnevich, Application of the concept of regular systems of points in metric number theory. (Russian) Vestsī Nats. Akad. Navuk Belarusī Ser. Fīz. -Mat. Navuk 2000, no. 1, 35–39.

  4. V. Beresnevich, D. Dickinson, and S. Velani, Measure theoretic laws for lim sup sets, Mem. Amer. Math. Soc. 179 (2006), no. 846.

    Google Scholar 

  5. V. Beresnevich, D. Dickinson, and S. Velani, Diophantine approximation on planar curves and the distribution of rational points. Ann. of Math. (2) 166 (2007), 367–426.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Boshernitzan and C. Carroll, An extension of Lagrange’s theorem to interval exchange trans-formations over quadratic fields. J. Anal. Math. 72 (1997), 21–44.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Boshernitzan and J. Chaika, Diophantine properties of IETs and general systems: quantitative proximality and connectivity, preprint.

  8. J. Cassels, Some metrical theorems in Diophantine approximation. I. Proc. Cambridge Philos. Soc. 46 (1950), 209–218.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Chaika, Shrinking targets for IETs: extending a theorem of Kurzweil, Geom. Funct. Anal. 21 (2011), 1020–1042.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. David, Wavelets and Singular Integrals on Curves and Surfaces, Springer Lecture Notes in Math 1465, 1991.

  11. G. David and S. Semmes, Fractured Fractals and Broken Dreams, Oxford University Press, 1997.

  12. F. Durand, B. Host, and C. Skau, Substitutional dynamical systems, Bratteli diagrams and dimension groups. Ergodic Theory Dynam. Systems 19 (1999), 953–993.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Elkies and C. McMullen, Gaps in \(\sqrt n \) mod 1 and ergodic theory, Duke Math. J. 123 (2004), 95–139.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Fayad, Mixing in the absence of the shrinking target property, Bull. London Math. Soc. 38 (2006), 829–838.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Heinonen, Lectures on Analysis on Metric Spaces, Springer, 2001.

  16. A. Khinchin, Continued Fractions, Dover, 1997.

  17. J. Kurzweil, On the metric theory of inhomogeneous diophantine approximations, Studia Math. 15 (1955), 84–112.

    MathSciNet  MATH  Google Scholar 

  18. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press, 1995.

  19. Z. Rudnick and P. Sarnak, The pair correlation function of fractional parts of polynomials, Comm. Math. Phys. 194 (1998), 61–70.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Tseng, On circle rotations and shrinking target properties, Discrete Contin. Dyn. Syst. 20 (2008), 1111–1122.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Veech, The metric theory of interval exchange transformations. Amer. J. Math. 106 (1984), 1331–1422.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    Michael Boshernitzan

  2. Department of Mathematics, University of Chicago, 5734 S. University, Chicago, IL, 60637, USA

    Jon Chaika

Authors
  1. Michael Boshernitzan
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  2. Jon Chaika
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Correspondence to Michael Boshernitzan.

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Boshernitzan, M., Chaika, J. Borel-Cantelli sequences. JAMA 117, 321–345 (2012). https://doi.org/10.1007/s11854-012-0024-4

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  • DOI: https://doi.org/10.1007/s11854-012-0024-4

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