Can you hear the shape of a beatty sequence?

Abstract

Let K(x1,⋯,xd) be a polynomial. If you are not given the real numbers α1,α1, ⋯ αd, but are given the polynomial K and the sequence a n = K(⌊nα1⌋, ⌊nα 2⌋, ⋯ ⌊nαd⌋), can you deduce the values of αi ? 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 ⌊ ⌊ ⋯ ⌊ ⌊nα 1⌋ ⋯ αd-1⌋α d⌋)n=1∞. © 2010 Springer Science+Business Media, LLC.