Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups

Abstract:

For a real number $\alpha$, the floor function $\left \lfloor \alpha \right \rfloor$ is the integer part of $\alpha$. The sequence $\{ \left \lfloor {m\alpha } \right \rfloor :m = 1,2,3, \ldots \}$ is the Beatty sequence of $\alpha$. Identities are proved which express the sum of the iterated floor functional ${A^i}$ for $1 \leq i \leq n$, operating on a nonzero algebraic number $\alpha$ of degree $\leq n$, in terms of only ${A^1} = \left \lfloor {m\alpha } \right \rfloor ,m$ and a bounded term. Applications include discrete chaos (discrete dynamical systems), explicit construction of infinite nonchaotic subsequences of chaotic sequences, discrete order (identities), explicit construction of nontrivial Beatty subsequences, and certain arithmetical semigroups. (Beatty sequences have a large literature in combinatorics. They have also been used in nonperiodic tilings (quasicrystallography), periodic scheduling, computer vision (digital lines), and formal language theory.)