Can you hear the shape of a Beatty sequence?

TL;DR

This paper investigates whether the original real parameters of polynomial-generated Beatty sequences can be uniquely determined from the sequences themselves, under certain irrationality conditions and specific polynomial forms.

Contribution

It introduces conditions under which the parameters of polynomial Beatty sequences can be uniquely recovered from the sequences, extending understanding of their identifiability.

Findings

Identifies conditions for parameter recovery in polynomial Beatty sequences.

Shows limitations of parameter deduction without additional hypotheses.

Analyzes nested flooring sequences and their reconstructibility.

Abstract

Let K(x_1,...,x_d) be a polynomial. If you are not given the real numbers \alpha_1, \alpha_2, ...,\alpha_d, but are given the polynomial K and the sequence a_n=K(\floor{n\alpha_1},\floor{n\alpha_2},...,\floor{n\alpha_d}), can you deduce the values of \alpha_i? Not, it turns out, in general. But with additional irrationality hypotheses and certain polynomials, it is possible. We also consider the problem of deducing \alpha_i from the integer sequence with nested flooring (\floor{\floor{... \floor{\floor{n\alpha_1}\alpha_2}... \alpha_{d-1}}\alpha_d})_{n=1}^\infty.