Integers in number systems with positive and negative quadratic Pisot base
Accepted: 17 March 2014
We consider numeration systems with base β and − β, for quadratic Pisot numbers β and focus on comparing the combinatorial structure of the sets Zβ and Z− β of numbers with integer expansion in base β, resp. − β. Our main result is the comparison of languages of infinite words uβ and u− β coding the ordering of distances between consecutive β- and (− β)-integers. It turns out that for a class of roots β of x2 − mx − m, the languages coincide, while for other quadratic Pisot numbers the language of uβ can be identified only with the language of a morphic image of u− β. We also study the group structure of (− β)-integers.
Mathematics Subject Classification: 11K16 / 68R15
Key words: Quadratic Pisot numbers / beta-integers / negative base
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