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On finite $β$-expansions for the set of natural numbers
We present a study of the problem of finiteness of the $β$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $β$-expansion of 1 is not a non-decreasing sequence. We show a class of simple $β$-numbers which display a puzzling behavi
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