Skip to main content
SpringerLink
Log in

Sets of bounded discrepancy for multi-dimensional irrational rotation

  • Published:
Geometric and Functional Analysis Aims and scope Submit manuscript

Abstract

We study bounded remainder sets with respect to an irrational rotation of the d-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one.

First we extend to several dimensions the Hecke–Ostrowski result by constructing a class of d-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of “equidecomposability” to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for a polytope to be a bounded remainder set. In particular this yields a characterization of the convex bounded remainder polygons in two dimensions. The approach is used to obtain several other results as well.

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. U. Bolle. On multiple tiles in E 2. Intuitive geometry. Proceedings of the 3rd international conference held in Szeged, Hungary, from 2 to 7 September 1991. North-Holland, Amsterdam; János Bolyai Mathematical Society, Budapest (1994), pp. 39–43.

  2. Boltianski V.: Hilbert’s Third Problem. Wiley, New York (1978)

    Google Scholar 

  3. Cassels J.W.S.: An Introduction to the Geometry of Numbers. Springer, New York (1997)

    MATH  Google Scholar 

  4. C. De Concini, and C. Procesi. Topics in Hyperplane Arrangements, Polytopes and Box-Splines. Springer, New York (2011).

  5. P. Erdős. Problems and results on diophantine approximations. Composito Mathematica, 16 (1964), 52–65

  6. Ferenczi S.: Bounded remainder sets. Acta Arithmetica 4(61), 319–326 (1992)

    MathSciNet  Google Scholar 

  7. Furstenberg H., Keynes H., Shapiro L.: Prime flows in topological dynamics. Israel Journal of Mathematics 14, 26–38 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  8. W.H. Gottschalk, and G.A. Hedlund. Topological dynamics. Colloquium Publications of the American Mathematical Society (AMS), Vol. 36. American Mathematical Society (AMS), Providence, Vol. VIII (1955), 151 p.

  9. Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1978)

    MATH  Google Scholar 

  10. H. Hadwiger. Translationsinvariante, additive und schwachstetige Polyederfunktionale. Archiv der Mathematik, 3 (1952), 387–394 (German)

  11. H. Hadwiger. Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Die Grundlehren der Mathematischen Wissenschaften, Vol. 93. Springer-Verlag, Berlin, Vol. XIII (1957), 312 S.

  12. H. Hadwiger. Translative Zerlegungsgleichheit der Polyeder des gewöhnlichen Raumes. Journal fur die Reine und Angewandte Mathematik, 233 (1968), 200–212 (German)

  13. H. Hadwiger, and P. Glur. Zerlegungsgleichheit ebener Polygone. Elemente der Mathematik, 6 (1951), 97–106 (German)

  14. Halász G.: Remarks on the remainder in Birkhoff’s ergodic theorem. Acta Mathematica Academiae Scientiarum Hungaricae 28, 389–395 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Hartman. Colloquium Mathematicum, 1 (1947), 239–240 (French)

  16. A. Haynes, and H. Koivusalo. Constructing Bounded Remainder Sets and Cut-and-Project Sets Which are Bounded Distance to Lattices (2014). arXiv:1402.2125.

  17. E. Hecke. Über analytische Funktionen und die Verteilung von Zahlen mod. eins. Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg, 1 (1921), 54–76 (German)

  18. B. Jessen. Zur Algebra der Polytope. Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl., II (1972), 47–53 (German)

  19. Jessen B., Thorup A.: The algebra of polytopes in affine spaces. Mathematica Scandinavica 43, 211–240 (1978)

    MATH  MathSciNet  Google Scholar 

  20. H. Kesten. On a conjecture of Erdős and Szüsz related to uniform distribution mod 1. Acta Arithmetica, 12 (1966), 193–212

  21. Liardet P.: Regularities of distribution. Composito Mathematica 61, 267–293 (1987)

    MATH  MathSciNet  Google Scholar 

  22. P. Mürner. Translative Zerlegungsgleichheit von Polytopen. Archiv der Mathematik, 29 (1977), 218–224 (German)

  23. Oren I.: Admissible functions with multiple discontinuities. Israel Journal of Mathematics 42, 353–360 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  24. A. Ostrowski. Mathematische Miszellen IX: Notiz zur Theorie der Diophantischen Approximationen. Jahresber Dtsch Math-Ver, 36 (1927), 178–180 (German)

  25. A. Ostrowski. Mathematische Miszellen. XVI: Zur Theorie der linearen diophantischen Approximationen. Jahresber Dtsch Math-Ver, 39 (1930), 34–46 (German)

  26. Petersen K.: On a series of cosecants related to a problem in ergodic theory. Composito Mathematica 26, 313–317 (1973)

    MATH  Google Scholar 

  27. G. Rauzy. Nombres algébriques et substitutions. Bulletin of the Society Mathematis of France, 110 (1982), 147–178 (French)

  28. G. Rauzy. Ensembles à à restes bornés. Sémin. Théor. Nombres, Univ. Bordeaux I 1983–1984, Exp. No. 24. (1984), 12 p. (French).

  29. C.-H. Sah. Hilbert’s third problem: scissors congruence. Research Notes in Mathematics, Vol. 33. San Francisco, London. Pitman Advanced Publishing Program, Vol. X (1979),188 p.

  30. J. Schoissengeier. Regularity of distribution of (nα)-sequences. Acta Arithmetica, (2)133 (2008), 127–157

  31. Shephard G.C.: Combinatorial properties of associated zonotopes. Canadian Journal of Mathematics, 26, 302–321 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  32. A.V. Shutov. On a family of two-dimensional bounded remainder sets. Chebyshevskiĭ Sb., (4 (40))12 (2011), 264–271 (Russian)

  33. J.-P. Sydler. Conditions nécessaires et suffisantes pour l’équivalence des polyèdres de l’espace euclidien à à trois dimensions. Commentarii Mathematici Helvetici, 40 (1965), 43–80 (French)

  34. P. Szüsz. Über die Verteilung der Vielfachen einer komplexen Zahl nach dem Modul des Einheitsquadrats. Acta Mathematica Academiae Scientiarum Hungaricae, 5 (1954), 35–39 (German)

  35. P. Szüsz. Lösung eines Problems von Herrn Hartman. Studia Mathematica, 15 (1955), 43–55 (German)

  36. Zhuravlev V.G.: Bounded remainder polyhedra. Proceedings of the Steklov Institute of Mathematics 280, 71–90 (2013)

    Article  MathSciNet  Google Scholar 

  37. Ziegler G.M.: Lectures on Polytopes. Springer-Verlag, Berlin (1995)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Sigrid Grepstad

  2. Department of Mathematics, Bar-Ilan University, Ramat-Gan, 52900, Tel Aviv, Israel

    Nir Lev

Authors
  1. Sigrid Grepstad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nir Lev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nir Lev.

Additional information

Research partially supported by the Israel Science Foundation grant No. 225/13.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Grepstad, S., Lev, N. Sets of bounded discrepancy for multi-dimensional irrational rotation. Geom. Funct. Anal. 25, 87–133 (2015). https://doi.org/10.1007/s00039-015-0313-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00039-015-0313-z

Keywords and phrases

Mathematical Subject Classification

Search

Navigation