## Généralisation d’un résultat de Loxton et van der Poorten. (Generalization of a result of Loxton and van der Poorten).(French)Zbl 0778.11015

This paper presents an interesting application of the theory of finite automata to a problem which arose in diophantine approximation. Let $$g$$ be an integer with $$g\geq 2$$ and let $$C$$ be a finite set of non-negative integers. The set $$C$$ is called $$g$$-free if it is not possible to find elements $$c_ i$$, $$c_ i'$$ in $$C$$ with $$\sum^ n_{i=0} c_ i g^ i=\sum^ n_{i=0} c_ i' g^ i$$ and $$(c_ 0,\dots,c_ n) \neq (c_ 0',\dots,c_ n')$$. It is relatively easy to see that if $$| C|>g$$ then $$C$$ cannot be $$g$$-free and if $$| C|<g$$ then there are arbitrarily large integers which cannot be written in the form $$\sum c_ k g^ k$$. Thus the most interesting case is $$| C|=g$$. The author uses the algebra of finite automata to characterise these sets $$C$$ which are $$g$$-free.
Suppose, without loss of generality, that 0 is in $$C$$. Let $$P(X)= \sum_{c\in C} X^ c$$ and $$P_ n(X)= \prod^{n-1}_{h=0} P(X^{g^ h})$$, let $$Z_ n$$ be the set of roots of $$P_ n$$ and let $$U_{g^ n}$$ be the set of $$g^ n$$-th roots of unity. Then $$C$$ is $$g$$-free if and only if $$| U_{g^ n} \setminus Z_ n|$$ is bounded independent of $$n$$.
The case $$g=4$$ and $$C=\{0,1,k,k+1\}$$ recovers a result of J. H. Loxton and A. J. van der Poorten [Acta Arith. 49, 193-203 (1987; Zbl 0636.10003)]. Suppose integers in base 4 are written using digits $$- 1$$, 0, 1 and 2. The given set $$C$$ is not 4-free if and only if there are integers $$s_ i$$, $$s_ i'$$ whose base 4 representations contains only the digits 0 and 1 such that $$s_ 2k+s_ 1=s_ 2' k+s_ 1'$$ and $$(s_ 1,s_ 2)\neq (s_ 1',s_ 2')$$. This equation gives $$k$$ as the quotient of two integers whose base 4 representations contain only the digits 0, 1 and $$-1$$. By the main theorem, the integers $$k$$ with this property are those $$k=4^ \alpha k'$$ with $$k'$$ odd.

### MSC:

 11B83 Special sequences and polynomials 68Q70 Algebraic theory of languages and automata 11A67 Other number representations

Zbl 0636.10003
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### References:

  Loxton, J.H. et Poorten, A. J. van der, An awful problem about integers in base 4, Acta Arith.49 (1987), 193-203. · Zbl 0636.10003  Rauzy, G., Systèmes de numération, Journées de théorie élémentaire et analytique des nombres, 1982, Valenciennes.
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