## Bases and ambiguity of number systems.(English)Zbl 0546.68066

A number system is a $$(v+1)-$$tuple $$N=(n,m_ 1,...,m_ v)$$ where $$n\geq 2,$$ $$v\geq 1$$ and $$m_ 1,...,m_ v\geq 1.$$ The word $$m_{i_ 0}...m_{i_ k}$$ represents the integer $$m_{i_ 0}\cdot n^ k+...+m_{i_ k}.$$ The set of represented integers is denoted by S(N). An integer b is called a base of S(N) if there exists a number system $$N_ 1=(b,a_ 1,...,a_ u)$$ such that $$S(N)=S(N_ 1)$$. We show that the bases of S(N) form a subfamily of an exponential family if S(N) is not a finite union of arithmetic progressions. If the set S(N) has arbitrarily long gaps, then S(N) has only one base. The degree of ambiguity of a number N is defined to be the greatest integer p such that some integer has p representations and none has more than p representations. If there is no such integer the degree of N is $$\infty$$. The degree of S(N) equals q if there is a number system M of degree q such that $$S(N)=S(M)$$ and there is no number system M’ of degree lower than q such that $$S(N)=S(M')$$. We show that all degrees of ambiguity do exist and that the degree of a given number system can be computed effectively.

### MSC:

 68Q45 Formal languages and automata 11A63 Radix representation; digital problems 68Q42 Grammars and rewriting systems

### Keywords:

number system; represented integers; base; degree of ambiguity
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### References:

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