Advertisement

Can You Hear the Shape of a Beatty Sequence?

  • Ron GrahamEmail author
  • Kevin O’Bryant
Chapter

Summary

Let \(K({x}_{1},\ldots,{x}_{d})\) be a polynomial. If you are not given the real numbers \({\alpha }_{1},{\alpha }_{2},\ldots,{\alpha }_{d}\), but are given the polynomial K and the sequence \({a}_{n} = K(\lfloor n{\alpha }_{1}\rfloor,\lfloor n{\alpha }_{2}\rfloor,\ldots,\lfloor n{\alpha }_{d}\rfloor )\), can you deduce the values of α i ? No, it turns out, in general. But with additional irrationality hypotheses and certain polynomials, it is possible. We also consider the problem of deducing α i from the integer sequence \({(\lfloor \lfloor \cdots \lfloor \lfloor n{\alpha }_{1}\rfloor {\alpha }_{2}\rfloor \cdots {\alpha }_{d-1}\rfloor {\alpha }_{d}\rfloor )}_{n=1}^{\infty }\).

Keywords

Beatty sequence Generalized polynomial 

Notes

Acknowledgements

We thank Inger Håland Knutson for helpful comments and nice examples. This work was supported (in part) by a grant from The City University of New York PSC-CUNY Research Award Program.

References

  1. 1.
    Vitaly Bergelson and Alexander Leibman, Distribution of values of bounded generalized polynomials, Acta Math. 198 (2007), no. 2, 155–230. MR 2318563 Google Scholar
  2. 2.
    Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete mathematics, 2nd ed. Addison-Wesley Publishing Company, Reading, MA, 1994. A foundation for computer science. MR 1397498 Google Scholar
  3. 3.
    Vitaly Bergelson, Inger J. Håland Knutson, and Randall McCutcheon, IP-systems, generalized polynomials and recurrence, Ergodic Theory Dynam. Systems, 26, (2006), no. 4, 999–1019. MR 2246589 Google Scholar
  4. 4.
    Inger Johanne Håland, Uniform distribution of generalized polynomials of the product type, Acta Arith., 67 (1994), no. 1, 13–27 MR 1292518 Google Scholar
  5. 5.
    Kevin O’Bryant, Fraenkel’s partition and Brown’s decomposition, Integers, 3 (2003), A11, 17 pp. (electronic). MR 2006610 Google Scholar
  6. 6.
    Kenneth Valbjørn Rasmussen, Ligefordelte følger i [0, 1] k med anvendelser, FAMøS 18 (2004), no. 2, 35–42.Google Scholar
  7. 7.
    I. M. Vinogradov, The method of trigonometrical sums in the theory of numbers, Dover Publications Inc. Mineola, NY, 2004. Translated from the Russian, revised and annotated by K. F. Roth and Anne Davenport; Reprint of the 1954 translation. MR 2104806 Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of CaliforniaSan DiegoUSA
  2. 2.College of Staten Island and Graduate CenterThe City University of New YorkNew YorkUSA

Personalised recommendations