# Can You Hear the Shape of a Beatty Sequence?

• Ron Graham
• 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.

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