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MR1138949 (94d:11011) 11B83 (11B39 11Z50 58F13)
Fraenkel, Aviezri S. Iterated floor function, algebraic numbers, discrete chaos, Beatty subsequences, semigroups. Trans. Amer. Math. Soc. 341 (1994), no. 2, 639–664.
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