Can You Hear the Shape of a Beatty Sequence?

Ron Graham () and Kevin O’Bryant ()
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Ron Graham: University of California
Kevin O’Bryant: The City University of New York, College of Staten Island and Graduate Center

A chapter in Additive Number Theory, 2010, pp 39-52 from Springer

Abstract: 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 (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/978-0-387-68361-4_3

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