Can You Hear the Shape of a Beatty Sequence?

  • Ron GrahamEmail author
  • Kevin O’Bryant


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 \({(\lfloor \lfloor \cdots \lfloor \lfloor n{\alpha }_{1}\rfloor {\alpha }_{2}\rfloor \cdots {\alpha }_{d-1}\rfloor {\alpha }_{d}\rfloor )}_{n=1}^{\infty }\).


Beatty sequence Generalized polynomial 



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.


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© 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

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