Skip to main content
SpringerLink
Log in

Good points for diophantine approximation

  • Published:
Proceedings - Mathematical Sciences Aims and scope Submit manuscript

Abstract

Given a sequence (x n ) ∞n=1 of real numbers in the interval [0, 1) and a sequence (δ n ) ∞n=1 of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x n | < δ n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x n ) ∞n=1 , which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ n ) ∞n=1 converges to zero. On the other hand, for any sequence of positive numbers (δ n ) ∞n=1 tending to zero, there is a well distributed sequence (x n ) ∞n=1 in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cassels J W S, An Introduction to Diophantine Approximation (Cambridge University Press) (1957)

  2. Dai X-R, Feng D-J and Wang Y, Refinable functions with non-integer dilations, J. Funct. Anal. 250 (2007) 1–20

    Article  MATH  MathSciNet  Google Scholar 

  3. Falconer K, Fractal Geometry — Mathematical Foundations and Applications, 2nd ed. (Chichester: Wiley) (2005)

    Google Scholar 

  4. Hlawka E, Zur formalen Theorie der Gleichverteilung in kompakten Gruppen, Rend. Circ. Math. Palermo (2) 4 (1955) 33–47

    Article  MATH  MathSciNet  Google Scholar 

  5. Kahane J-P, Some Random Series of Functions, 2nd ed. (Cambridge: Cambridge University Press) (1985)

    MATH  Google Scholar 

  6. Kuipers L and Niederreiter H, Uniform Distribution of Sequences (New York: Wiley-Interscience) (1974)

    MATH  Google Scholar 

  7. Mitrinović DS, Sándor J and Crstici B, Handbook of Number Theory (Dordrecht, Boston: Kluwer) (1996)

    Google Scholar 

  8. Petersen G M, Almost convergence and uniformly distributed sequences, Quart. J. Math. (2) 7 (1956) 189–191

    Google Scholar 

  9. Ridout D, Rational approximations to algebraic numbers, Mathematika 4 (1957) 125–131

    Article  MATH  MathSciNet  Google Scholar 

  10. Shepp L A, Covering the circle with random arcs, Israel J. Math. 11 (1972) 328–345

    Article  MATH  MathSciNet  Google Scholar 

  11. Strauch O and Porubský Š, Distribution of Sequences: A Sampler, Schriftenreihe der Slowakischen Akademie der Wissenschaften 1 (Frankfurt: Peter Lang) (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva, 84105, Israel

    Daniel Berend

  2. Department of Mathematics and Informatics, Vilnius University, Vilnius, LT-03225, Lithuania

    Artūras Dubickas

Authors
  1. Daniel Berend
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Artūras Dubickas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Berend.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berend, D., Dubickas, A. Good points for diophantine approximation. Proc Math Sci 119, 423–429 (2009). https://doi.org/10.1007/s12044-009-0040-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12044-009-0040-1

Keywords

Search

Navigation