Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Aviezri S. Fraenkel
Journal: Trans. Amer. Math. Soc. 341 (1994), 639-664
MSC: Primary 11B83; Secondary 11B39, 11Z50, 58F13
MathSciNet review: 1138949
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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.)

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Article copyright: © Copyright 1994 American Mathematical Society