Minimal weight expansions in Pisot bases
Résumé
For numeration systems representing real numbers and integers, which are based on Pisot numbers, we study expansions with signed digits which are minimal with respect to the absolute sum of digits. It is proved that these expansions are recognizable by a finite automaton if the base $\beta$ is a Pisot number satisfying a certain condition (D$'$). When $\beta$ is the Golden Ratio, the Tribonacci number or the smallest Pisot number, the digits are $-1,0,1$, and the automaton is given explicitly. We compute the average number of non-zero digits for these representations, which is lower than for the celebrated non-adjacent form (in base 2).
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