Addition and multiplication of beta-expansions in generalized Tribonacci base
Petr Ambrož, Zuzana Masáková, Edita Pelantová
Abstract
We study properties of β-numeration systems,
where β > 1 is the real root of the polynomial
x3 - mx2 - x - 1,
m ∈ ℕ,
m ≥ 1.
We consider arithmetic operations on the set of
β-integers, i.e., on the set of numbers whose greedy
expansion in base β has no fractional part.
We show that the number of fractional digits arising
under addition of β-integers is at most 5 for
m ≥ 3 and 6 for m = 2, whereas under
multiplication it is at most 6 for all m ≥ 2.
We thus generalize the results known for Tribonacci
numeration system, i.e., for m = 1.
We summarize the combinatorial properties of infinite
words naturally defined by β-integers. We point
out the differences between the structure
of β-integers in cases m = 1 and m ≥ 2.
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